# What is Linear Algebra? - A First Course in Linear Algebra

We begin our study of linear algebra with an introduction and a motivational example.

• ##### Definition of linear equation math.la.d.lineqn
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# Solving Systems of Linear Equations - A First Course in Linear Algebra

We will motivate our study of linear algebra by considering the problem of solving several linear equations simultaneously. The word solve tends to get abused somewhat, as in “solve this problem.” When talking about equations we understand a more precise meaning: find all of the values of some variable quantities that make an equation, or several equations, simultaneously true.

• ##### Definition of system of linear equations math.la.d.linsys
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6 years ago
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# Solving Systems of Linear Equations - A First Course in Linear Algebra

We will motivate our study of linear algebra by considering the problem of solving several linear equations simultaneously. The word solve tends to get abused somewhat, as in “solve this problem.” When talking about equations we understand a more precise meaning: find all of the values of some variable quantities that make an equation, or several equations, simultaneously true.

• ##### Definition of solution set of a system of linear equations math.la.d.linsys.soln_set
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September 11th, 2017
6 years ago
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# Solving Systems of Linear Equations - A First Course in Linear Algebra

We will motivate our study of linear algebra by considering the problem of solving several linear equations simultaneously. The word solve tends to get abused somewhat, as in “solve this problem.” When talking about equations we understand a more precise meaning: find all of the values of some variable quantities that make an equation, or several equations, simultaneously true.

• ##### Definition of solution to a system of linear equations math.la.d.linsys.soln
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September 11th, 2017
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# Using matrices to solve linear systems

Equivalence of systems of linear equations, row operations, corresponding matrices representing the linear systems

• ##### Example of putting a matrix in echelon form math.la.e.mat.echelon.of
Created On
August 21st, 2017
6 years ago
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# Solving Systems of Linear Equations - A First Course in Linear Algebra

We will motivate our study of linear algebra by considering the problem of solving several linear equations simultaneously. The word solve tends to get abused somewhat, as in “solve this problem.” When talking about equations we understand a more precise meaning: find all of the values of some variable quantities that make an equation, or several equations, simultaneously true.

• ##### Definition of equivalent systems of linear equations math.la.d.linsys.equiv
GFDL-1.2
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September 11th, 2017
6 years ago
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# Solving Systems of Linear Equations - A First Course in Linear Algebra

We will motivate our study of linear algebra by considering the problem of solving several linear equations simultaneously. The word solve tends to get abused somewhat, as in “solve this problem.” When talking about equations we understand a more precise meaning: find all of the values of some variable quantities that make an equation, or several equations, simultaneously true.

• ##### Equation operations on a linear system give an equivalent system. math.la.t.linsys.op
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September 11th, 2017
6 years ago
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# Solving Systems of Linear Equations - A First Course in Linear Algebra

We will motivate our study of linear algebra by considering the problem of solving several linear equations simultaneously. The word solve tends to get abused somewhat, as in “solve this problem.” When talking about equations we understand a more precise meaning: find all of the values of some variable quantities that make an equation, or several equations, simultaneously true.

• ##### Definition of equation operations on a linear system math.la.d.linsys.op
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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# Notation for matrix entries, size of a matrix, etc

Notation for matrix entries, diagonal matrix, square matrix, identity matrix, and zero matrix.

• ##### Definition of zero matrix math.la.d.mat.zero
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# Vector Arithmetic

Definition of vector, equality of vectors, vector addition, and scalar vector multiplication. Geometric and algebraic properties of vector addition are discussed. (need a topic on vector addition is commutative and associative)

• ##### math.la.t.vec.sum.geometric.RnCn
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February 19th, 2017
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# Vector Addition and Scalar Multiplication

University of Waterloo Math Online -

• ##### Definition of linear combination of vectors, coordinate vector space math.la.d.vec.lincomb.coord
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9 years ago
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# Vector Addition and Scalar Multiplication

Slides for the accompanying video from University of Waterloo.

• ##### Definition of scalar, coordinate vector space math.la.d.scalar
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9 years ago
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# Reduced Row-Echelon Form - A First Course in Linear Algebra

After solving a few systems of equations, you will recognize that it does not matter so much what we call our variables, as opposed to what numbers act as their coefficients. A system in the variables $$x_1,\,x_2,\,x_3$$ would behave the same if we changed the names of the variables to $$a,\,b,\,c$$ and kept all the constants the same and in the same places. In this section, we will isolate the key bits of information about a system of equations into something called a matrix, and then use this matrix to systematically solve the equations. Along the way we will obtain one of our most important and useful computational tools.

• ##### math.la.c.mat.entry
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September 11th, 2017
6 years ago
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# Reduced Row-Echelon Form - A First Course in Linear Algebra

After solving a few systems of equations, you will recognize that it does not matter so much what we call our variables, as opposed to what numbers act as their coefficients. A system in the variables $$x_1,\,x_2,\,x_3$$ would behave the same if we changed the names of the variables to $$a,\,b,\,c$$ and kept all the constants the same and in the same places. In this section, we will isolate the key bits of information about a system of equations into something called a matrix, and then use this matrix to systematically solve the equations. Along the way we will obtain one of our most important and useful computational tools.

• ##### Definition of entry/component of a vector, coordinate vector space math.la.d.vec.component.coord
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6 years ago
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# Reduced Row-Echelon Form - A First Course in Linear Algebra

After solving a few systems of equations, you will recognize that it does not matter so much what we call our variables, as opposed to what numbers act as their coefficients. A system in the variables $$x_1,\,x_2,\,x_3$$ would behave the same if we changed the names of the variables to $$a,\,b,\,c$$ and kept all the constants the same and in the same places. In this section, we will isolate the key bits of information about a system of equations into something called a matrix, and then use this matrix to systematically solve the equations. Along the way we will obtain one of our most important and useful computational tools.

• ##### math.la.d.mat.constant
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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# Reduced Row-Echelon Form - A First Course in Linear Algebra

After solving a few systems of equations, you will recognize that it does not matter so much what we call our variables, as opposed to what numbers act as their coefficients. A system in the variables $$x_1,\,x_2,\,x_3$$ would behave the same if we changed the names of the variables to $$a,\,b,\,c$$ and kept all the constants the same and in the same places. In this section, we will isolate the key bits of information about a system of equations into something called a matrix, and then use this matrix to systematically solve the equations. Along the way we will obtain one of our most important and useful computational tools.

• ##### Definition of coefficient matrix of a linear system math.la.d.mat.coeff
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6 years ago
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# Reduced Row-Echelon Form - A First Course in Linear Algebra

After solving a few systems of equations, you will recognize that it does not matter so much what we call our variables, as opposed to what numbers act as their coefficients. A system in the variables $$x_1,\,x_2,\,x_3$$ would behave the same if we changed the names of the variables to $$a,\,b,\,c$$ and kept all the constants the same and in the same places. In this section, we will isolate the key bits of information about a system of equations into something called a matrix, and then use this matrix to systematically solve the equations. Along the way we will obtain one of our most important and useful computational tools.

• ##### Definition of augmented matrix of a linear system math.la.d.mat.augmented
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September 11th, 2017
6 years ago
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# Reduced Row-Echelon Form - A First Course in Linear Algebra

After solving a few systems of equations, you will recognize that it does not matter so much what we call our variables, as opposed to what numbers act as their coefficients. A system in the variables $$x_1,\,x_2,\,x_3$$ would behave the same if we changed the names of the variables to $$a,\,b,\,c$$ and kept all the constants the same and in the same places. In this section, we will isolate the key bits of information about a system of equations into something called a matrix, and then use this matrix to systematically solve the equations. Along the way we will obtain one of our most important and useful computational tools.

• ##### Definition of solution vector of a linear system math.la.d.vec.solution
GFDL-1.2
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September 11th, 2017
6 years ago
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# Reduced Row-Echelon Form - A First Course in Linear Algebra

After solving a few systems of equations, you will recognize that it does not matter so much what we call our variables, as opposed to what numbers act as their coefficients. A system in the variables $$x_1,\,x_2,\,x_3$$ would behave the same if we changed the names of the variables to $$a,\,b,\,c$$ and kept all the constants the same and in the same places. In this section, we will isolate the key bits of information about a system of equations into something called a matrix, and then use this matrix to systematically solve the equations. Along the way we will obtain one of our most important and useful computational tools.

• ##### Definition of zero vector, coordinate vector space math.la.d.vec.z.coord
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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# Reduced Row-Echelon Form - A First Course in Linear Algebra

After solving a few systems of equations, you will recognize that it does not matter so much what we call our variables, as opposed to what numbers act as their coefficients. A system in the variables $$x_1,\,x_2,\,x_3$$ would behave the same if we changed the names of the variables to $$a,\,b,\,c$$ and kept all the constants the same and in the same places. In this section, we will isolate the key bits of information about a system of equations into something called a matrix, and then use this matrix to systematically solve the equations. Along the way we will obtain one of our most important and useful computational tools.

• ##### Definition of matrix representation of a linear system math.la.d.linsys.mat.repn
GFDL-1.2
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6 years ago
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# Using matrices to solve linear systems

Equivalence of systems of linear equations, row operations, corresponding matrices representing the linear systems

• ##### Example of putting a matrix in echelon form math.la.e.mat.echelon.of
Created On
August 21st, 2017
6 years ago
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# Reduced Row-Echelon Form - A First Course in Linear Algebra

After solving a few systems of equations, you will recognize that it does not matter so much what we call our variables, as opposed to what numbers act as their coefficients. A system in the variables $$x_1,\,x_2,\,x_3$$ would behave the same if we changed the names of the variables to $$a,\,b,\,c$$ and kept all the constants the same and in the same places. In this section, we will isolate the key bits of information about a system of equations into something called a matrix, and then use this matrix to systematically solve the equations. Along the way we will obtain one of our most important and useful computational tools.

• ##### Definition of row operations on a matrix math.la.d.mat.row_op
GFDL-1.2
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September 11th, 2017
6 years ago
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# Reduced Row-Echelon Form - A First Course in Linear Algebra

After solving a few systems of equations, you will recognize that it does not matter so much what we call our variables, as opposed to what numbers act as their coefficients. A system in the variables $$x_1,\,x_2,\,x_3$$ would behave the same if we changed the names of the variables to $$a,\,b,\,c$$ and kept all the constants the same and in the same places. In this section, we will isolate the key bits of information about a system of equations into something called a matrix, and then use this matrix to systematically solve the equations. Along the way we will obtain one of our most important and useful computational tools.

• ##### Row equivalent matrices represent equivalent linear systems math.la.t.mat.row_equiv.linsys
GFDL-1.2
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6 years ago
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# When does a linear system have a unique solution?

A 3x3 system having a unique solution is solved by putting the augmented matrix in reduced row echelon form. A picture of three intersecting planes provides geometric intuition.

• ##### Definition of consistent linear system math.la.d.linsys.consistent
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# Echelon form

Definition of echelon form, reduction of a matrix to echelon form in order to compute solutions to systems of linear equations; definition of reduced row echelon form

• ##### Example of putting a matrix in echelon form math.la.e.mat.echelon.of
Created On
August 25th, 2017
6 years ago
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# Reduced Row-Echelon Form - A First Course in Linear Algebra

After solving a few systems of equations, you will recognize that it does not matter so much what we call our variables, as opposed to what numbers act as their coefficients. A system in the variables $$x_1,\,x_2,\,x_3$$ would behave the same if we changed the names of the variables to $$a,\,b,\,c$$ and kept all the constants the same and in the same places. In this section, we will isolate the key bits of information about a system of equations into something called a matrix, and then use this matrix to systematically solve the equations. Along the way we will obtain one of our most important and useful computational tools.

• ##### Every matrix is row-equivalent to a matrix in reduced row echelon form. math.la.t.mat.rref.exists
GFDL-1.2
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September 11th, 2017
6 years ago
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# Reduced Row-Echelon Form - A First Course in Linear Algebra

After solving a few systems of equations, you will recognize that it does not matter so much what we call our variables, as opposed to what numbers act as their coefficients. A system in the variables $$x_1,\,x_2,\,x_3$$ would behave the same if we changed the names of the variables to $$a,\,b,\,c$$ and kept all the constants the same and in the same places. In this section, we will isolate the key bits of information about a system of equations into something called a matrix, and then use this matrix to systematically solve the equations. Along the way we will obtain one of our most important and useful computational tools.

• ##### Every matrix is row-equivalent to only one matrix in reduced row echelon form. math.la.t.mat.rref.unique
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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# Reduced Row-Echelon Form - A First Course in Linear Algebra

After solving a few systems of equations, you will recognize that it does not matter so much what we call our variables, as opposed to what numbers act as their coefficients. A system in the variables $$x_1,\,x_2,\,x_3$$ would behave the same if we changed the names of the variables to $$a,\,b,\,c$$ and kept all the constants the same and in the same places. In this section, we will isolate the key bits of information about a system of equations into something called a matrix, and then use this matrix to systematically solve the equations. Along the way we will obtain one of our most important and useful computational tools.

• ##### Definition of leading entry in a row of a matrix math.la.d.mat.row.leading
GFDL-1.2
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September 11th, 2017
6 years ago
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# When does a linear system have a unique solution?

A 3x3 system having a unique solution is solved by putting the augmented matrix in reduced row echelon form. A picture of three intersecting planes provides geometric intuition.

• ##### Definition of consistent linear system math.la.d.linsys.consistent
Created On
February 15th, 2017
6 years ago
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3
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# Solution Sets

Sample problems to help understand when a linear system has 0, 1, or infinitely many solutions.

• ##### math.la.t.rref.consistent
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February 15th, 2017
6 years ago
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# Types of Solution Sets - A First Course in Linear Algebra

We will now be more careful about analyzing the reduced row-echelon form derived from the augmented matrix of a system of linear equations. In particular, we will see how to systematically handle the situation when we have infinitely many solutions to a system, and we will prove that every system of linear equations has either zero, one or infinitely many solutions. With these tools, we will be able to routinely solve any linear system.

• ##### Definition of free/independent variable in a linear system math.la.d.linsys.variable.independent
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6 years ago
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# Types of Solution Sets - A First Course in Linear Algebra

We will now be more careful about analyzing the reduced row-echelon form derived from the augmented matrix of a system of linear equations. In particular, we will see how to systematically handle the situation when we have infinitely many solutions to a system, and we will prove that every system of linear equations has either zero, one or infinitely many solutions. With these tools, we will be able to routinely solve any linear system.

• ##### math.la.t.rref.pivot
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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Textbook
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# Types of Solution Sets - A First Course in Linear Algebra

We will now be more careful about analyzing the reduced row-echelon form derived from the augmented matrix of a system of linear equations. In particular, we will see how to systematically handle the situation when we have infinitely many solutions to a system, and we will prove that every system of linear equations has either zero, one or infinitely many solutions. With these tools, we will be able to routinely solve any linear system.

• ##### math.la.t.rref.consistent
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
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# Types of Solution Sets - A First Course in Linear Algebra

We will now be more careful about analyzing the reduced row-echelon form derived from the augmented matrix of a system of linear equations. In particular, we will see how to systematically handle the situation when we have infinitely many solutions to a system, and we will prove that every system of linear equations has either zero, one or infinitely many solutions. With these tools, we will be able to routinely solve any linear system.

• ##### Definition of inconsistent linear system math.la.d.linsys.inconsistent
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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# When does a linear system have a unique solution?

A 3x3 system having a unique solution is solved by putting the augmented matrix in reduced row echelon form. A picture of three intersecting planes provides geometric intuition.

• ##### Definition of consistent linear system math.la.d.linsys.consistent
Created On
February 15th, 2017
6 years ago
Views
3
Type
Video
Timeframe
Review
Perspective
Example
Language
English
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# Solution Sets

Sample problems to help understand when a linear system has 0, 1, or infinitely many solutions.

• ##### math.la.t.rref.consistent
Created On
February 15th, 2017
6 years ago
Views
2
Type
Handout
Timeframe
In-class
Perspective
Example
Language
English
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# Types of Solution Sets - A First Course in Linear Algebra

We will now be more careful about analyzing the reduced row-echelon form derived from the augmented matrix of a system of linear equations. In particular, we will see how to systematically handle the situation when we have infinitely many solutions to a system, and we will prove that every system of linear equations has either zero, one or infinitely many solutions. With these tools, we will be able to routinely solve any linear system.

• ##### Linear systems have zero, one, or infinitely many solutions. math.la.t.linsys.zoi
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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Textbook
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# Types of Solution Sets - A First Course in Linear Algebra

We will now be more careful about analyzing the reduced row-echelon form derived from the augmented matrix of a system of linear equations. In particular, we will see how to systematically handle the situation when we have infinitely many solutions to a system, and we will prove that every system of linear equations has either zero, one or infinitely many solutions. With these tools, we will be able to routinely solve any linear system.

• ##### The number of pivots in the reduced row echelon form of a consistent system determines the number of free variables in the solution set. math.la.t.rref.pivot.free
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
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Textbook
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# Types of Solution Sets - A First Course in Linear Algebra

We will now be more careful about analyzing the reduced row-echelon form derived from the augmented matrix of a system of linear equations. In particular, we will see how to systematically handle the situation when we have infinitely many solutions to a system, and we will prove that every system of linear equations has either zero, one or infinitely many solutions. With these tools, we will be able to routinely solve any linear system.

• ##### A consistent system with more variables than equations has infinitely many solutions. math.la.t.linsys.consistent.i
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
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Textbook
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# Finding all solutions of systems of linear equations

How to compute all solutions to a general system $Ax=b$ of linear equations and connection to the corresponding homogeneous system $Ax=0$. Visualization of the geometry of solution sets. Consistent systems and their solution using row reduction.

• ##### Definition of nontrivial solution to a homogeneous linear system of equations math.la.d.linsys.homog.nontrivial
Created On
August 22nd, 2017
6 years ago
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# Solving homogeneous systems of equations

Homogeneous systems of linear equations; trivial versus nontrivial solutions of homogeneous systems; how to find nontrivial solutions; how to know from the reduced row-echelon form of a matrix whether the corresponding homogeneous system has nontrivial solutions.

• ##### Example of solving a 3-by-3 homogeneous system of linear equations by row-reducing the augmented matrix, in the case of infinitely many solutions math.la.e.linsys.3x3.soln.homog.row_reduce.i
Created On
August 25th, 2017
6 years ago
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4
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# Homogeneous Systems of Equations - A First Course in Linear Algebra

In this section we specialize to systems of linear equations where every equation has a zero as its constant term. Along the way, we will begin to express more and more ideas in the language of matrices and begin a move away from writing out whole systems of equations. The ideas initiated in this section will carry through the remainder of the course.

• ##### Definition of homogeneous linear system of equations math.la.d.linsys.homog
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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# Homogeneous Systems of Equations - A First Course in Linear Algebra

In this section we specialize to systems of linear equations where every equation has a zero as its constant term. Along the way, we will begin to express more and more ideas in the language of matrices and begin a move away from writing out whole systems of equations. The ideas initiated in this section will carry through the remainder of the course.

• ##### Homogeneous linear systems are consistent. math.la.d.linsys.homog.consistent
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
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Textbook
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# Homogeneous Systems of Equations - A First Course in Linear Algebra

In this section we specialize to systems of linear equations where every equation has a zero as its constant term. Along the way, we will begin to express more and more ideas in the language of matrices and begin a move away from writing out whole systems of equations. The ideas initiated in this section will carry through the remainder of the course.

• ##### A homog system with more variables than equations has infinitely many solutions. math.la.t.linsys.homog.i
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
Type
Textbook
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English
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# Homogeneous Systems of Equations - A First Course in Linear Algebra

In this section we specialize to systems of linear equations where every equation has a zero as its constant term. Along the way, we will begin to express more and more ideas in the language of matrices and begin a move away from writing out whole systems of equations. The ideas initiated in this section will carry through the remainder of the course.

• ##### Definition of trivial solution to a homogeneous linear system of equations math.la.d.linsys.homog.trivial
GFDL-1.2
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September 11th, 2017
6 years ago
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2
Type
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# Homogeneous Systems of Equations - A First Course in Linear Algebra

In this section we specialize to systems of linear equations where every equation has a zero as its constant term. Along the way, we will begin to express more and more ideas in the language of matrices and begin a move away from writing out whole systems of equations. The ideas initiated in this section will carry through the remainder of the course.

• ##### Definition of matrix null space (right) math.la.d.mat.null_space.right
GFDL-1.2
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September 11th, 2017
6 years ago
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# The Matrix Inverse - Definition

Motivation and definition of the inverse of a matrix

• ##### Matrix inverses are unique: if A and B are square matrices, then AB=I implies that A=B^-1 and B=A^-1. math.la.t.mat.inv.unique
(CC-BY-NC-SA-4.0 OR CC-BY-SA-4.0)
Created On
January 5th, 2017
6 years ago
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3
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# Notation for matrix entries, size of a matrix, etc

Notation for matrix entries, diagonal matrix, square matrix, identity matrix, and zero matrix.

• ##### Definition of zero matrix math.la.d.mat.zero
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6 years ago
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# Matrix Inverse

The definition of matrix inverse is motivated by considering multiplicative inverse. The identity matrix and matrix inverse are defined.

• ##### The identity matrix is the identity for matrix multiplication. math.la.t.mat.mult.identity
Created On
February 17th, 2017
6 years ago
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2
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Video
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Introduction
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# Invertible Matrix Theorem

Statements that are equivalent to a square matrix being invertible; examples.

• ##### Definition of matrix inverse math.la.d.mat.inv
Created On
August 21st, 2017
6 years ago
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3
Type
Video
Language
English
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# Matrix inverses

Properties of matrix inversion: inverse of the inverse, inverse of the transpose, inverse of a product; elementary matrices and corresponding row operations; a matrix is invertible if and only if it is row-equivalent to the identity matrix; row-reduction algorithm for computing matrix inverse

• ##### Equivalence theorem: the matrix A row-reduces to the identity matrix. math.la.t.equiv.identity
Created On
August 25th, 2017
6 years ago
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3
Type
Video
Language
English
Content Type
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# Nonsingular Matrices - A First Course in Linear Algebra

In this section we specialize further and consider matrices with equal numbers of rows and columns, which when considered as coefficient matrices lead to systems with equal numbers of equations and variables. We will see in the second half of the course (Chapter D, Chapter E, Chapter LT, Chapter R) that these matrices are especially important.

• ##### Definition of singular matrix math.la.d.mat.singular
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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Textbook
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# Nonsingular Matrices - A First Course in Linear Algebra

In this section we specialize further and consider matrices with equal numbers of rows and columns, which when considered as coefficient matrices lead to systems with equal numbers of equations and variables. We will see in the second half of the course (Chapter D, Chapter E, Chapter LT, Chapter R) that these matrices are especially important.

• ##### Equivalence theorem: the matrix A row-reduces to the identity matrix. math.la.t.equiv.identity
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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2
Type
Textbook
Language
English
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# Nonsingular Matrices - A First Course in Linear Algebra

In this section we specialize further and consider matrices with equal numbers of rows and columns, which when considered as coefficient matrices lead to systems with equal numbers of equations and variables. We will see in the second half of the course (Chapter D, Chapter E, Chapter LT, Chapter R) that these matrices are especially important.

• ##### Definition of square matrix math.la.d.mat.square
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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2
Type
Textbook
Language
English
Content Type
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# Nonsingular Matrices - A First Course in Linear Algebra

In this section we specialize further and consider matrices with equal numbers of rows and columns, which when considered as coefficient matrices lead to systems with equal numbers of equations and variables. We will see in the second half of the course (Chapter D, Chapter E, Chapter LT, Chapter R) that these matrices are especially important.

• ##### Definition of identity matrix math.la.d.mat.identity
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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2
Type
Textbook
Language
English
Content Type
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# Nonsingular Matrices - A First Course in Linear Algebra

In this section we specialize further and consider matrices with equal numbers of rows and columns, which when considered as coefficient matrices lead to systems with equal numbers of equations and variables. We will see in the second half of the course (Chapter D, Chapter E, Chapter LT, Chapter R) that these matrices are especially important.

• ##### Equivalence theorem: the equation Ax=b has a solution for all b. math.la.t.equiv.mat.eqn.unique
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# Nonsingular Matrices - A First Course in Linear Algebra

In this section we specialize further and consider matrices with equal numbers of rows and columns, which when considered as coefficient matrices lead to systems with equal numbers of equations and variables. We will see in the second half of the course (Chapter D, Chapter E, Chapter LT, Chapter R) that these matrices are especially important.

• ##### math.la.t.equiv.mat.eqn.unique.rep
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

# Nonsingular Matrices - A First Course in Linear Algebra

In this section we specialize further and consider matrices with equal numbers of rows and columns, which when considered as coefficient matrices lead to systems with equal numbers of equations and variables. We will see in the second half of the course (Chapter D, Chapter E, Chapter LT, Chapter R) that these matrices are especially important.

• ##### Equivalence theorem: the null space of the matrix A is {0}. math.la.t.equiv.nullspace
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
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# Vector Arithmetic

Definition of vector, equality of vectors, vector addition, and scalar vector multiplication. Geometric and algebraic properties of vector addition are discussed. (need a topic on vector addition is commutative and associative)

• ##### math.la.t.vec.sum.geometric.RnCn
Created On
February 19th, 2017
6 years ago
Views
2
Type
Video
Timeframe
Pre-class
Perspective
Introduction
Language
English
Content Type
text/html; charset=utf-8

# Vector Addition and Scalar Multiplication

University of Waterloo Math Online -

• ##### Definition of linear combination of vectors, coordinate vector space math.la.d.vec.lincomb.coord
Created On
October 23rd, 2013
9 years ago
Views
2
Type
Video
Perspective
Introduction
Language
English
Content Type
text/html;charset=UTF-8

# Vector Addition and Scalar Multiplication

Slides for the accompanying video from University of Waterloo.

• ##### Definition of scalar, coordinate vector space math.la.d.scalar
Created On
October 23rd, 2013
9 years ago
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3
Type
Handout
Perspective
Introduction
Language
English
Content Type
application/pdf

# Vector Addition and Scalar Multiplication Quiz

Quiz from the University of Waterloo.

• ##### Theorem: a set of vectors is linearly dependent if and only if one of the vectors can be written as a linear combination of the other vectors, coordinate vector space. math.la.t.vec.lindep.coord
Created On
October 23rd, 2013
9 years ago
Views
4
Type
Unknown
Timeframe
Post-class
Perspective
Example
Language
English
Content Type
text/html;charset=UTF-8

# Vector Operations - A First Course in Linear Algebra

In this section we define some new operations involving vectors, and collect some basic properties of these operations. Begin by recalling our definition of a column vector as an ordered list of complex numbers, written vertically (Definition CV). The collection of all possible vectors of a fixed size is a commonly used set, so we start with its definition.

• ##### Definition of vector sum/addition, coordinate vector space math.la.d.vec.sum.coord
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
Type
Textbook
Language
English
Content Type
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# Vector Operations - A First Course in Linear Algebra

In this section we define some new operations involving vectors, and collect some basic properties of these operations. Begin by recalling our definition of a column vector as an ordered list of complex numbers, written vertically (Definition CV). The collection of all possible vectors of a fixed size is a commonly used set, so we start with its definition.

• ##### math.la.d.vec.rncn
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# Vector Operations - A First Course in Linear Algebra

In this section we define some new operations involving vectors, and collect some basic properties of these operations. Begin by recalling our definition of a column vector as an ordered list of complex numbers, written vertically (Definition CV). The collection of all possible vectors of a fixed size is a commonly used set, so we start with its definition.

• ##### Definition of vector-scalar multiplication, coordinate vector space math.la.d.vec.scalar.mult.coord
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# Vector Operations - A First Course in Linear Algebra

In this section we define some new operations involving vectors, and collect some basic properties of these operations. Begin by recalling our definition of a column vector as an ordered list of complex numbers, written vertically (Definition CV). The collection of all possible vectors of a fixed size is a commonly used set, so we start with its definition.

• ##### Definition of equality of vectors, coordinate vector space math.la.d.vec.equal.coord
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# Vector Operations - A First Course in Linear Algebra

In this section we define some new operations involving vectors, and collect some basic properties of these operations. Begin by recalling our definition of a column vector as an ordered list of complex numbers, written vertically (Definition CV). The collection of all possible vectors of a fixed size is a commonly used set, so we start with its definition.

• ##### math.la.t.vec.sum.coord
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

# Matrix equations as motivation for basis and span

Advice to instructors for in-class activities on matrix-vector multiplication and translating between the various equivalent notation forms of linear systems, and suggestions for how this topic can be used to motivate future topics.

• ##### The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A. math.la.t.mat.eqn.lincomb
Created On
February 15th, 2017
6 years ago
Views
3
Type
Handout
Timeframe
In-class
Perspective
Example
Language
English
Content Type
text/html; charset=utf-8

# Matrix equations with zero or infinitely many solutions

A 3x3 matrix equation Ax=b is solved for two different values of b. In one case there is no solution, and in another there are infinitely many solutions. These examples illustrate a theorem about linear combinations of the columns of the matrix A.

• ##### Definition of matrix equation math.la.d.mat.eqn
Created On
February 15th, 2017
6 years ago
Views
3
Type
Video
Timeframe
Pre-class
Perspective
Example
Language
English
Content Type
text/html; charset=utf-8

# Linear combinations of vectors

The linear combination of a set of vectors is defined. Determine if a vector in R^2 is in the span of two other vectors. The span of a set of vectors is related to the columns of a matrix. (need topic: Determine if a vector in R^2 is in the span of two other vectors.)

• ##### The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A. math.la.t.mat.eqn.lincomb
Created On
February 20th, 2017
6 years ago
Views
2
Type
Video
Timeframe
Pre-class
Perspective
Introduction
Language
English
Content Type
text/html; charset=utf-8

# Span of a set of vectors

Definition of the span of a set of vectors. Example of checking if a vector in R^3 is in the span of a set of two vectors. Geometric picture of a span.

• ##### Geometric description of span of a set of vectors in R^n (or C^n) math.la.c.vec.span.geometric.rncn
Created On
February 20th, 2017
6 years ago
Views
3
Type
Video
Timeframe
Pre-class
Perspective
Introduction
Language
English
Content Type
text/html; charset=utf-8

# Vector Equations: In-Class Activities

Suggestions for in-class activities on linear combination and span of vectors in R^n. (need a topic for the general *process* of determining if a vector is in the span of a set of devtors)

• ##### A linear system is equivalent to a vector equation. math.la.t.linsys.vec
Created On
February 20th, 2017
6 years ago
Views
2
Type
Handout
Timeframe
In-class
Language
English
Content Type
text/html; charset=utf-8

# Linear Combinations and Span

In-class activity for linear combinations and span.

GFDL-1.3
Created On
June 8th, 2017
6 years ago
Views
3
Type
Handout
Timeframe
In-class
Perspective
Introduction
Language
English
Content Type
application/pdf

# Vector Addition and Scalar Multiplication

University of Waterloo Math Online -

• ##### Definition of linear combination of vectors, coordinate vector space math.la.d.vec.lincomb.coord
Created On
October 23rd, 2013
9 years ago
Views
2
Type
Video
Perspective
Introduction
Language
English
Content Type
text/html;charset=UTF-8

# Matrix equations and spanning sets

Equivalent statements for a matrix A: for every right-hand side b, the system Ax=b has a solution; every b is a linear combination of the columns of A; the span of the columns of A is maximal; A has a pivot position in every row.

• ##### The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A. math.la.t.mat.eqn.lincomb
Created On
August 25th, 2017
6 years ago
Views
2
Type
Video
Language
English
Content Type
text/html; charset=utf-8

# Linear Combinations - A First Course in Linear Algebra

In Section VO we defined vector addition and scalar multiplication. These two operations combine nicely to give us a construction known as a linear combination, a construct that we will work with throughout this course.

• ##### The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A. math.la.t.mat.eqn.lincomb
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# Linear Combinations - A First Course in Linear Algebra

In Section VO we defined vector addition and scalar multiplication. These two operations combine nicely to give us a construction known as a linear combination, a construct that we will work with throughout this course.

• ##### Definition of linear combination of vectors, coordinate vector space math.la.d.vec.lincomb.coord
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

# Column and Row Spaces - A First Course in Linear Algebra

A matrix-vector product (Definition MVP) is a linear combination of the columns of the matrix and this allows us to connect matrix multiplication with systems of equations via Theorem SLSLC. Row operations are linear combinations of the rows of a matrix, and of course, reduced row-echelon form (Definition RREF) is also intimately related to solving systems of equations. In this section we will formalize these ideas with two key definitions of sets of vectors derived from a matrix.

• ##### The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A. math.la.t.mat.eqn.lincomb
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

# Linear Combinations - A First Course in Linear Algebra

In Section VO we defined vector addition and scalar multiplication. These two operations combine nicely to give us a construction known as a linear combination, a construct that we will work with throughout this course.

• ##### math.la.t.linsys.soln.vector
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# Linear Combinations - A First Course in Linear Algebra

In Section VO we defined vector addition and scalar multiplication. These two operations combine nicely to give us a construction known as a linear combination, a construct that we will work with throughout this course.

• ##### The solutions to a nonhomogeneous system are given by a particular solution plus the solutions to the homogeneous system. math.la.t.linsys.nonhomog.particular_plus_homog
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# Span of a set of vectors

Definition of the span of a set of vectors. Example of checking if a vector in R^3 is in the span of a set of two vectors. Geometric picture of a span.

• ##### Geometric description of span of a set of vectors in R^n (or C^n) math.la.c.vec.span.geometric.rncn
Created On
February 20th, 2017
6 years ago
Views
3
Type
Video
Timeframe
Pre-class
Perspective
Introduction
Language
English
Content Type
text/html; charset=utf-8

# Vector Equations: In-Class Activities

Suggestions for in-class activities on linear combination and span of vectors in R^n. (need a topic for the general *process* of determining if a vector is in the span of a set of devtors)

• ##### A linear system is equivalent to a vector equation. math.la.t.linsys.vec
Created On
February 20th, 2017
6 years ago
Views
2
Type
Handout
Timeframe
In-class
Language
English
Content Type
text/html; charset=utf-8

# Linear Combinations and Span

In-class activity for linear combinations and span.

GFDL-1.3
Created On
June 8th, 2017
6 years ago
Views
3
Type
Handout
Timeframe
In-class
Perspective
Introduction
Language
English
Content Type
application/pdf

# Properties of Vectors and Spanning

From the University of Waterloo Math Online

• ##### Geometric description of span of a set of vectors in R^n (or C^n) math.la.c.vec.span.geometric.rncn
Created On
October 23rd, 2013
9 years ago
Views
3
Type
Video
Perspective
Introduction
Language
English
Content Type
text/html;charset=UTF-8

# Properties of Vectors and Spanning

Slides from the corresponding video from the University of Waterloo.

• ##### Geometric description of span of a set of vectors in R^n (or C^n) math.la.c.vec.span.geometric.rncn
Created On
October 23rd, 2013
9 years ago
Views
3
Type
Handout
Perspective
Introduction
Language
English
Content Type
application/pdf

# Spanning Sets - A First Course in Linear Algebra

In this section we will provide an extremely compact way to describe an infinite set of vectors, making use of linear combinations. This will give us a convenient way to describe the solution set of a linear system, the null space of a matrix, and many other sets of vectors.

• ##### Definition of span of a set of vectors, coordinate vector space math.la.d.vec.span.coord
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# Span of a set of vectors

Definition of the span of a set of vectors. Example of checking if a vector in R^3 is in the span of a set of two vectors. Geometric picture of a span.

• ##### Geometric description of span of a set of vectors in R^n (or C^n) math.la.c.vec.span.geometric.rncn
Created On
February 20th, 2017
6 years ago
Views
3
Type
Video
Timeframe
Pre-class
Perspective
Introduction
Language
English
Content Type
text/html; charset=utf-8

# Vector Equations: In-Class Activities

Suggestions for in-class activities on linear combination and span of vectors in R^n. (need a topic for the general *process* of determining if a vector is in the span of a set of devtors)

• ##### A linear system is equivalent to a vector equation. math.la.t.linsys.vec
Created On
February 20th, 2017
6 years ago
Views
2
Type
Handout
Timeframe
In-class
Language
English
Content Type
text/html; charset=utf-8

# Linear Combinations and Span

In-class activity for linear combinations and span.

GFDL-1.3
Created On
June 8th, 2017
6 years ago
Views
3
Type
Handout
Timeframe
In-class
Perspective
Introduction
Language
English
Content Type
application/pdf

# Properties of Vectors and Spanning

From the University of Waterloo Math Online

• ##### Geometric description of span of a set of vectors in R^n (or C^n) math.la.c.vec.span.geometric.rncn
Created On
October 23rd, 2013
9 years ago
Views
3
Type
Video
Perspective
Introduction
Language
English
Content Type
text/html;charset=UTF-8

# Properties of Vectors and Spanning

Slides from the corresponding video from the University of Waterloo.

• ##### Geometric description of span of a set of vectors in R^n (or C^n) math.la.c.vec.span.geometric.rncn
Created On
October 23rd, 2013
9 years ago
Views
3
Type
Handout
Perspective
Introduction
Language
English
Content Type
application/pdf

# Spanning Sets - A First Course in Linear Algebra

In this section we will provide an extremely compact way to describe an infinite set of vectors, making use of linear combinations. This will give us a convenient way to describe the solution set of a linear system, the null space of a matrix, and many other sets of vectors.

• ##### Definition of span of a set of vectors, coordinate vector space math.la.d.vec.span.coord
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# Spanning Sets - A First Course in Linear Algebra

In this section we will provide an extremely compact way to describe an infinite set of vectors, making use of linear combinations. This will give us a convenient way to describe the solution set of a linear system, the null space of a matrix, and many other sets of vectors.

• ##### math.la.t.mat.null_space.rref.span
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# How to find out if a set of vectors is linearly independent?

Linear independence is defined, followed by a worked example of 3 vectors in R^3.

• ##### Definition of linearly independent set of vectors: if a linear combination is zero, then every coefficient is zero, coordinate vector space. math.la.d.vec.linindep.coord
Created On
February 15th, 2017
6 years ago
Views
3
Type
Video
Timeframe
Pre-class
Perspective
Introduction
Language
English
Content Type
text/html; charset=utf-8

# Linear Independence in-class activity

Linear independence in-class activity

GFDL-1.3
Created On
June 8th, 2017
6 years ago
Views
3
Type
Handout
Timeframe
In-class
Perspective
Example
Language
English
Content Type
application/pdf

# University of Waterloo Math Online - Lesson: Linear Independence and Surfaces

Video Lesson from University of Waterloo.

• ##### Theorem: a set of vectors is linearly dependent if and only if one of the vectors can be written as a linear combination of the other vectors, coordinate vector space. math.la.t.vec.lindep.coord
Created On
October 23rd, 2013
9 years ago
Views
2
Type
Video
Perspective
Introduction
Language
English
Content Type
text/html;charset=UTF-8

# Vector Addition and Scalar Multiplication Quiz

Quiz from the University of Waterloo.

• ##### Theorem: a set of vectors is linearly dependent if and only if one of the vectors can be written as a linear combination of the other vectors, coordinate vector space. math.la.t.vec.lindep.coord
Created On
October 23rd, 2013
9 years ago
Views
4
Type
Unknown
Timeframe
Post-class
Perspective
Example
Language
English
Content Type
text/html;charset=UTF-8

# Linear Independence - A First Course in Linear Algebra

Linear independence is one of the most fundamental conceptual ideas in linear algebra, along with the notion of a span. So this section, and the subsequent Section LDS, will explore this new idea.

• ##### math.la.d.vec.lindep.relation.trivial
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

# Linear Independence - A First Course in Linear Algebra

Linear independence is one of the most fundamental conceptual ideas in linear algebra, along with the notion of a span. So this section, and the subsequent Section LDS, will explore this new idea.

• ##### Definition of linearly independent set of vectors: if a linear combination is zero, then every coefficient is zero, coordinate vector space. math.la.d.vec.linindep.coord
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

# Linear Independence - A First Course in Linear Algebra

Linear independence is one of the most fundamental conceptual ideas in linear algebra, along with the notion of a span. So this section, and the subsequent Section LDS, will explore this new idea.

• ##### If a set of vectors in R^n (or C^n) contains more than n elements, then the set is linearly dependent. math.la.t.vec.lindep.more.rncn
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

# Linear Independence - A First Course in Linear Algebra

Linear independence is one of the most fundamental conceptual ideas in linear algebra, along with the notion of a span. So this section, and the subsequent Section LDS, will explore this new idea.

• ##### math.la.t.vec.linindep.pivot
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# Linear Independence - A First Course in Linear Algebra

Linear independence is one of the most fundamental conceptual ideas in linear algebra, along with the notion of a span. So this section, and the subsequent Section LDS, will explore this new idea.

• ##### math.la.t.vec.linindep.homog
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# Linear Independence - A First Course in Linear Algebra

Linear independence is one of the most fundamental conceptual ideas in linear algebra, along with the notion of a span. So this section, and the subsequent Section LDS, will explore this new idea.

• ##### Equivalence theorem: the columns of A are linearly independent. math.la.t.equiv.col.linindep
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# Linear Independence - A First Course in Linear Algebra

Linear independence is one of the most fundamental conceptual ideas in linear algebra, along with the notion of a span. So this section, and the subsequent Section LDS, will explore this new idea.

• ##### math.la.t.equiv.col.linindep.rep
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# Linear Independence - A First Course in Linear Algebra

Linear independence is one of the most fundamental conceptual ideas in linear algebra, along with the notion of a span. So this section, and the subsequent Section LDS, will explore this new idea.

• ##### math.la.t.mat.null_space.rref.basis
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

# Linear Independence in-class activity

Linear independence in-class activity

GFDL-1.3
Created On
June 8th, 2017
6 years ago
Views
3
Type
Handout
Timeframe
In-class
Perspective
Example
Language
English
Content Type
application/pdf

# University of Waterloo Math Online - Lesson: Linear Independence and Surfaces

Video Lesson from University of Waterloo.

• ##### Theorem: a set of vectors is linearly dependent if and only if one of the vectors can be written as a linear combination of the other vectors, coordinate vector space. math.la.t.vec.lindep.coord
Created On
October 23rd, 2013
9 years ago
Views
2
Type
Video
Perspective
Introduction
Language
English
Content Type
text/html;charset=UTF-8

# Vector Addition and Scalar Multiplication Quiz

Quiz from the University of Waterloo.

• ##### Theorem: a set of vectors is linearly dependent if and only if one of the vectors can be written as a linear combination of the other vectors, coordinate vector space. math.la.t.vec.lindep.coord
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October 23rd, 2013
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# Linear Dependence and Spans - A First Course in Linear Algebra

In any linearly dependent set there is always one vector that can be written as a linear combination of the others. This is the substance of the upcoming Theorem DLDS. Perhaps this will explain the use of the word “dependent.” In a linearly dependent set, at least one vector “depends” on the others (via a linear combination).

• ##### Theorem: a set of vectors is linearly dependent if and only if one of the vectors can be written as a linear combination of the other vectors, coordinate vector space. math.la.t.vec.lindep.coord
GFDL-1.2
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# Linear Dependence and Spans - A First Course in Linear Algebra

In any linearly dependent set there is always one vector that can be written as a linear combination of the others. This is the substance of the upcoming Theorem DLDS. Perhaps this will explain the use of the word “dependent.” In a linearly dependent set, at least one vector “depends” on the others (via a linear combination).

• ##### A set of nonzero vectors contains (as a subset) a basis for its span. math.la.t.vsp.span.basis
GFDL-1.2
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# Orthogonality - A First Course in Linear Algebra

In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use $$\complexes$$ as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers $${\mathbb R}\text{.}$$ If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in $$\complex{m}\text{.}$$

• ##### math.la.d.vec.conjugate.cn
GFDL-1.2
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6 years ago
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# Orthogonality - A First Course in Linear Algebra

In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use $$\complexes$$ as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers $${\mathbb R}\text{.}$$ If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in $$\complex{m}\text{.}$$

• ##### math.la.t.vec.sum.conjugate.cn
GFDL-1.2
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6 years ago
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2
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# Orthogonality - A First Course in Linear Algebra

In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use $$\complexes$$ as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers $${\mathbb R}\text{.}$$ If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in $$\complex{m}\text{.}$$

• ##### math.la.t.vec.scalar.mult.conjugate.cn
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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# Orthogonality - A First Course in Linear Algebra

In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use $$\complexes$$ as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers $${\mathbb R}\text{.}$$ If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in $$\complex{m}\text{.}$$

• ##### math.la.t.innerproduct.commutative.scalar.cn
GFDL-1.2
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6 years ago
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3
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# Orthogonality - A First Course in Linear Algebra

In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use $$\complexes$$ as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers $${\mathbb R}\text{.}$$ If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in $$\complex{m}\text{.}$$

• ##### math.la.t.innerproduct.distributive.rncn
GFDL-1.2
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September 11th, 2017
6 years ago
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# Orthogonality - A First Course in Linear Algebra

In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use $$\complexes$$ as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers $${\mathbb R}\text{.}$$ If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in $$\complex{m}\text{.}$$

• ##### math.la.t.innerproduct.commutative.cn
GFDL-1.2
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6 years ago
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2
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# Orthogonality - A First Course in Linear Algebra

In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use $$\complexes$$ as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers $${\mathbb R}\text{.}$$ If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in $$\complex{m}\text{.}$$

• ##### math.la.d.innerproduct.cn
GFDL-1.2
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6 years ago
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3
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# Dot Product and Cross Product Lesson

This is a video from the University of Waterloo. Dot Product, Cross-Product in R^n (which should be in Chapter 8 section 4 about hyperplanes.

• ##### Definition of cross product math.la.d.crossproduct
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October 23rd, 2013
9 years ago
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# Dot Product, Cross Product, and Scalar Equations Quiz

Quiz from the University of Waterloo. This is intended to be used after the video of the same name.

• ##### Definition of two vectors being orthogonal math.la.d.vec.orthogonal
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October 23rd, 2013
9 years ago
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# Inner products and distance

Inner product of two vectors in R^n, length of a vector in R^n, orthogonality. Motivation via approximate solutions of systems of linear equations, definition and properties of inner product (symmetric, bilinar, positive definite); length/norm of a vector, unit vectors; definition of distance between vectors; definition of orthogonality; Pythagorean Theorem.

• ##### Two vectors are orthogonal if and only if the Pythagorean Theorem holds. math.la.t.vec.orthogonal
Created On
August 22nd, 2017
6 years ago
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# Orthogonality - A First Course in Linear Algebra

In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use $$\complexes$$ as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers $${\mathbb R}\text{.}$$ If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in $$\complex{m}\text{.}$$

• ##### Definition of norm/length of a vector, coordinate setting math.la.d.vec.norm.coord
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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# Orthogonality - A First Course in Linear Algebra

In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use $$\complexes$$ as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers $${\mathbb R}\text{.}$$ If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in $$\complex{m}\text{.}$$

• ##### The standard inner product of a vector with itself is zero only for the zero vector, coordinate setting. math.la.t.innerproduct.self.z.coord
GFDL-1.2
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6 years ago
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2
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# Orthogonality - A First Course in Linear Algebra

In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use $$\complexes$$ as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers $${\mathbb R}\text{.}$$ If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in $$\complex{m}\text{.}$$

• ##### math.la.t.vec.innerproduct.norm
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
Type
Textbook
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Content Type
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# Dot Product and Cross Product Lesson

This is a video from the University of Waterloo. Dot Product, Cross-Product in R^n (which should be in Chapter 8 section 4 about hyperplanes.

• ##### Definition of cross product math.la.d.crossproduct
Created On
October 23rd, 2013
9 years ago
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3
Type
Video
Perspective
Introduction
Language
English
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# Dot Product, Cross Product, and Scalar Equations Quiz

Quiz from the University of Waterloo. This is intended to be used after the video of the same name.

• ##### Definition of two vectors being orthogonal math.la.d.vec.orthogonal
Created On
October 23rd, 2013
9 years ago
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3
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# Inner products and distance

Inner product of two vectors in R^n, length of a vector in R^n, orthogonality. Motivation via approximate solutions of systems of linear equations, definition and properties of inner product (symmetric, bilinar, positive definite); length/norm of a vector, unit vectors; definition of distance between vectors; definition of orthogonality; Pythagorean Theorem.

• ##### Two vectors are orthogonal if and only if the Pythagorean Theorem holds. math.la.t.vec.orthogonal
Created On
August 22nd, 2017
6 years ago
Views
2
Type
Video
Language
English
Content Type
text/html; charset=utf-8

# Orthogonality - A First Course in Linear Algebra

In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use $$\complexes$$ as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers $${\mathbb R}\text{.}$$ If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in $$\complex{m}\text{.}$$

• ##### An orthogonal set of nonzero vectors is linearly independent. math.la.t.vec.orthogonal_set.linindep
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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Textbook
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# Orthogonality - A First Course in Linear Algebra

In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use $$\complexes$$ as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers $${\mathbb R}\text{.}$$ If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in $$\complex{m}\text{.}$$

• ##### math.la.d.vec.orthogonal.coord
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
Type
Textbook
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English
Content Type
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# Orthogonality - A First Course in Linear Algebra

In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use $$\complexes$$ as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers $${\mathbb R}\text{.}$$ If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in $$\complex{m}\text{.}$$

• ##### Definition of orthogonal set of vectors math.la.d.vec.orthogonal_set
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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# Orthogonality - A First Course in Linear Algebra

In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use $$\complexes$$ as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers $${\mathbb R}\text{.}$$ If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in $$\complex{m}\text{.}$$

• ##### Definition of unit vector, coordinate setting math.la.d.vec.unit.coord
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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# Orthonormality

Orthonormal sets and bases (definition); expressing vectors as linear combinations of orthonormal basis vectors; matrices with orthonormal columns preserve vector norm and dot product; orthogonal matrices; inverse of an orthogonal matrix equals its transpose

• ##### A matrix A with real entries has orthonormal columns if and only if A inverse equals A transpose. math.la.t.mat.col.orthonormal.inv.rn
Created On
August 25th, 2017
6 years ago
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3
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Video
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# Orthogonality - A First Course in Linear Algebra

In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use $$\complexes$$ as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers $${\mathbb R}\text{.}$$ If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in $$\complex{m}\text{.}$$

• ##### Definition of orthonormal set of vectors math.la.d.vec.orthonormal_set
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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# Orthogonality - A First Course in Linear Algebra

In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use $$\complexes$$ as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers $${\mathbb R}\text{.}$$ If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in $$\complex{m}\text{.}$$

• ##### The Gram-Schmidt process converts a basis into an orthogonal basis. math.la.t.gramschmidt
GFDL-1.2
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# Matrix Equation: Matrix-Vector Product

The product of a matrix times a vector is defined, and used to show that a system of linear equations is equivalent to a system of linear equations involving matrices and vectors. The example uses a 2x3 system.

• ##### A matrix equation is equivalent to a linear system math.la.t.mat.eqn.linsys
CC-BY-SA-4.0
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February 15th, 2017
6 years ago
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3
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# Matrix Operations: Sums Scalar Multiplication

Definition of sum of matrices, product of a scalar and a matrix

• ##### Matrix-scalar multiplication is commutative, associative, and distributive. math.la.t.mat.scalar.mult.commut_assoc
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6 years ago
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# In-class activities for matrix operations

Suggestions for in-class activities on matrix operations: addition, multiplication, transpose, and the fact that multiplication is not commutative.

• ##### For matrices, AB=AC does not imply B=C in general. math.la.c.mat.mult.cancellation
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February 17th, 2017
6 years ago
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# Matrix Operations - A First Course in Linear Algebra

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

• ##### Definition of matrix-scalar multiplication math.la.d.mat.scalar.mult
GFDL-1.2
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# Matrix Operations - A First Course in Linear Algebra

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

• ##### math.la.d.mat.m_by_n.set
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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# Matrix Operations - A First Course in Linear Algebra

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

• ##### Definition of sum of matrices math.la.d.mat.sum
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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# Matrix Operations - A First Course in Linear Algebra

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

• ##### Definition of equality of matrices math.la.d.mat.equal
GFDL-1.2
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September 11th, 2017
6 years ago
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# Matrix Operations - A First Course in Linear Algebra

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

• ##### math.la.d.mat.z
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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# Matrix Operations - A First Course in Linear Algebra

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

• ##### math.la.e.vsp.mat.m_by_n
GFDL-1.2
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September 11th, 2017
6 years ago
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# Vector Spaces - A First Course in Linear Algebra

In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

• ##### math.la.e.vsp.mat.m_by_n
GFDL-1.2
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September 11th, 2017
6 years ago
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# Definition and properties of matrix transpose

The transpose of a matrix is defined, and various properties are explored using numerical examples.

• ##### Matrix transpose is an involution. math.la.t.mat.transpose.involution
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February 17th, 2017
6 years ago
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# Matrix operations

Associative and distributive properties of matrix multiplication and addition; multiplication by the identity matrix; definition of the transpose of a matrix; transpose of the transpose, transpose of a sum, transpose of a product

• ##### The transpose of a product of matrices is the product of the transposes in reverse order. math.la.t.mat.mult.transpose
Created On
August 25th, 2017
6 years ago
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2
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Video
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# Matrix Operations - A First Course in Linear Algebra

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

• ##### Definition of transpose of a matrix math.la.d.mat.transpose
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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# Matrix Operations - A First Course in Linear Algebra

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

• ##### math.la.t.mat.sum.transpose
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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# Matrix Operations - A First Course in Linear Algebra

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

• ##### Matrix transpose is an involution. math.la.t.mat.transpose.involution
GFDL-1.2
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September 11th, 2017
6 years ago
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# Matrix Operations - A First Course in Linear Algebra

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

• ##### math.la.t.mat.symmetric.square
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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# Matrix Operations - A First Course in Linear Algebra

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

• ##### Definition of the diagonal of a matrix math.la.d.mat.thediagonal
GFDL-1.2
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6 years ago
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# Matrix Operations - A First Course in Linear Algebra

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

• ##### Definition of symmetric matrix math.la.d.mat.symmetric
GFDL-1.2
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# Matrix Operations - A First Course in Linear Algebra

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

• ##### math.la.t.mat.scalar.transpose
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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Textbook
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English
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# Matrix Operations - A First Course in Linear Algebra

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

• ##### math.la.t.mat.scalar.conjugate
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
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# Matrix Operations - A First Course in Linear Algebra

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

• ##### math.la.t.mat.conjugate.involution
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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# Matrix Operations - A First Course in Linear Algebra

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

• ##### math.la.d.mat.conjugate
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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# Matrix Operations - A First Course in Linear Algebra

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

• ##### math.la.t.matsum.conjugate
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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# Matrix Operations - A First Course in Linear Algebra

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

• ##### math.la.t.mat.transpose.conjugate
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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Textbook
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# Matrix Operations - A First Course in Linear Algebra

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
Type
Textbook
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English
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# Matrix Operations - A First Course in Linear Algebra

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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2
Type
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# Matrix Operations - A First Course in Linear Algebra

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

• ##### math.la.d.mat.skewsymmetric
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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English
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# Matrix Operations - A First Course in Linear Algebra

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
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# Matrix Operations - A First Course in Linear Algebra

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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2
Type
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English
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# Matrix Equation: Matrix-Vector Product

The product of a matrix times a vector is defined, and used to show that a system of linear equations is equivalent to a system of linear equations involving matrices and vectors. The example uses a 2x3 system.

• ##### A matrix equation is equivalent to a linear system math.la.t.mat.eqn.linsys
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February 15th, 2017
6 years ago
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3
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# Matrix equations as motivation for basis and span

Advice to instructors for in-class activities on matrix-vector multiplication and translating between the various equivalent notation forms of linear systems, and suggestions for how this topic can be used to motivate future topics.

• ##### The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A. math.la.t.mat.eqn.lincomb
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6 years ago
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# How to multiply matrices

Learning goals: 1. What are the dimension (size) requirements for two matrices so that they can be multiplied to each other? 2. What is the product of two matrices, when it exists?

• ##### Definition of matrix-vector product, as a linear combination of column vectors math.la.d.mat.vec.prod
Created On
February 17th, 2017
6 years ago
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2
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# Matrix Multiplication - A First Course in Linear Algebra

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

• ##### math.la.t.mat.vec.prod.unique
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# Matrix Multiplication - A First Course in Linear Algebra

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

• ##### Definition of matrix-vector product, as a linear combination of column vectors math.la.d.mat.vec.prod
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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2
Type
Textbook
Language
English
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# Matrix Multiplication - A First Course in Linear Algebra

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

• ##### A matrix equation is equivalent to a linear system math.la.t.mat.eqn.linsys
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# How to multiply matrices

Learning goals: 1. What are the dimension (size) requirements for two matrices so that they can be multiplied to each other? 2. What is the product of two matrices, when it exists?

• ##### Definition of matrix-vector product, as a linear combination of column vectors math.la.d.mat.vec.prod
Created On
February 17th, 2017
6 years ago
Views
2
Type
Video
Timeframe
Review
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English
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# Matrix Multiplication - A First Course in Linear Algebra

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

• ##### Definition of matrix multiplication in terms of column vectors math.la.d.mat.mult.col
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
Type
Textbook
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English
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text/html

# Matrix Multiplication - A First Course in Linear Algebra

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

• ##### math.la.c.mat.mult
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

# Matrix Multiplication - A First Course in Linear Algebra

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

• ##### math.la.t.mat.mult.coord
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
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# Definition and properties of matrix transpose

The transpose of a matrix is defined, and various properties are explored using numerical examples.

• ##### Matrix transpose is an involution. math.la.t.mat.transpose.involution
Created On
February 17th, 2017
6 years ago
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2
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# Matrix Inverse

The definition of matrix inverse is motivated by considering multiplicative inverse. The identity matrix and matrix inverse are defined.

• ##### The identity matrix is the identity for matrix multiplication. math.la.t.mat.mult.identity
Created On
February 17th, 2017
6 years ago
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2
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# Matrix operations

Associative and distributive properties of matrix multiplication and addition; multiplication by the identity matrix; definition of the transpose of a matrix; transpose of the transpose, transpose of a sum, transpose of a product

• ##### The transpose of a product of matrices is the product of the transposes in reverse order. math.la.t.mat.mult.transpose
Created On
August 25th, 2017
6 years ago
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2
Type
Video
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English
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# Matrix Multiplication - A First Course in Linear Algebra

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

• ##### math.la.t.mat.mult.conjugate
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# Matrix Multiplication - A First Course in Linear Algebra

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

• ##### Matrix multiplication is associative. math.la.t.mat.mult.assoc
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# Matrix Multiplication - A First Course in Linear Algebra

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

• ##### The transpose of a product of matrices is the product of the transposes in reverse order. math.la.t.mat.mult.transpose
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# Matrix Multiplication - A First Course in Linear Algebra

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

• ##### Matrix multiplication is distributive over matrix addition. math.la.t.mat.mult.distributive
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
Type
Textbook
Language
English
Content Type
text/html

# Matrix Multiplication - A First Course in Linear Algebra

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

• ##### math.la.t.mat.mult.z
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# Matrix Multiplication - A First Course in Linear Algebra

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

• ##### math.la.t.innerproduct.mat.cn
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

# Matrix Multiplication - A First Course in Linear Algebra

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

• ##### The identity matrix is the identity for matrix multiplication. math.la.t.mat.mult.identity
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

# Matrix Multiplication - A First Course in Linear Algebra

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# Matrix Multiplication - A First Course in Linear Algebra

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

• ##### Matrix-scalar product is commutative math.la.t.mat.scalar.prod.commut
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# Matrix Multiplication - A First Course in Linear Algebra

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

• ##### Definition of Hermitian matrix math.la.d.mat.hermitian
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
Type
Textbook
Language
English
Content Type
text/html

# Matrix Multiplication - A First Course in Linear Algebra

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# Matrix Multiplication - A First Course in Linear Algebra

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

• ##### math.la.d.mat.hermitian.innerproduct.cn
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# The Matrix Inverse - Definition

Motivation and definition of the inverse of a matrix

• ##### Matrix inverses are unique: if A and B are square matrices, then AB=I implies that A=B^-1 and B=A^-1. math.la.t.mat.inv.unique
(CC-BY-NC-SA-4.0 OR CC-BY-SA-4.0)
Created On
January 5th, 2017
6 years ago
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3
Type
Video
Timeframe
Pre-class
Perspective
Introduction
Language
English
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# Matrix Inverse

The definition of matrix inverse is motivated by considering multiplicative inverse. The identity matrix and matrix inverse are defined.

• ##### The identity matrix is the identity for matrix multiplication. math.la.t.mat.mult.identity
Created On
February 17th, 2017
6 years ago
Views
2
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Video
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Pre-class
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Introduction
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English
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# Inverse of a Matrix: In-Class Activities

Suggested classroom activities on matrix inverses.

• ##### The inverse of a matrix (if it exists) can be found by row reducing the matrix augmented by the identity matrix. math.la.t.mat.inv.augmented
Created On
February 19th, 2017
6 years ago
Views
2
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Handout
Timeframe
In-class
Perspective
Introduction
Language
English
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# Lagrange Interpolation - Ximera

This is a guided discovery of the formula for Lagrange Interpolation, which lets you find the formula for a polynomial which passes through a given set of points.

• ##### Definition of linear transformation, arbitrary vector space math.la.d.lintrans.arb
Created On
June 8th, 2017
6 years ago
Views
2
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Handout
Perspective
Application
Language
English
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# Invertible Matrix Theorem

Statements that are equivalent to a square matrix being invertible; examples.

• ##### Definition of matrix inverse math.la.d.mat.inv
Created On
August 21st, 2017
6 years ago
Views
3
Type
Video
Language
English
Content Type
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# Matrix Inverses, definition and examples

Definition of the inverse of a matrix, examples, uniqueness; formula for the inverse of a 2x2 matrix; determinant of a 2x2 matrix; using the inverse to solve a system of linear equations.

• ##### The inverse of a matrix can be used to solve a linear system. math.la.t.eqn.mat.inv
Created On
August 22nd, 2017
6 years ago
Views
4
Type
Video
Language
English
Content Type
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# Matrix Inverses and Systems of Linear Equations - A First Course in Linear Algebra

The inverse of a square matrix, and solutions to linear systems with square coefficient matrices, are intimately connected.

• ##### math.la.d.mat.invertible
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

# Finding the inverse of a matrix and using it to solve a linear system

Matrix inverses are motivated as a way to solve a linear system. The general algorithm of finding an inverse by row reducing an augmented matrix is described, and then implemented for a 3x3 matrix. Useful facts about inverses are stated and then illustrated with sample 2x2 matrices. (put first: need Example of finding the inverse of a 3-by-3 matrix by row reducing the augmented matrix)

• ##### The inverse of a matrix can be used to solve a linear system. math.la.t.eqn.mat.inv
Created On
February 19th, 2017
6 years ago
Views
3
Type
Video
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Pre-class
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Introduction
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English
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# Inverse of a Matrix: In-Class Activities

Suggested classroom activities on matrix inverses.

• ##### The inverse of a matrix (if it exists) can be found by row reducing the matrix augmented by the identity matrix. math.la.t.mat.inv.augmented
Created On
February 19th, 2017
6 years ago
Views
2
Type
Handout
Timeframe
In-class
Perspective
Introduction
Language
English
Content Type
text/html; charset=utf-8

# Invertible Matrix Theorem

Statements that are equivalent to a square matrix being invertible; examples.

• ##### Definition of matrix inverse math.la.d.mat.inv
Created On
August 21st, 2017
6 years ago
Views
3
Type
Video
Language
English
Content Type
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# Matrix inverses

Properties of matrix inversion: inverse of the inverse, inverse of the transpose, inverse of a product; elementary matrices and corresponding row operations; a matrix is invertible if and only if it is row-equivalent to the identity matrix; row-reduction algorithm for computing matrix inverse

• ##### Equivalence theorem: the matrix A row-reduces to the identity matrix. math.la.t.equiv.identity
Created On
August 25th, 2017
6 years ago
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3
Type
Video
Language
English
Content Type
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# Matrix Inverses and Systems of Linear Equations - A First Course in Linear Algebra

The inverse of a square matrix, and solutions to linear systems with square coefficient matrices, are intimately connected.

• ##### Formula for the inverse of a 2-by-2 matrix. math.la.t.mat.inv.2x2
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

# Matrix Inverses and Systems of Linear Equations - A First Course in Linear Algebra

The inverse of a square matrix, and solutions to linear systems with square coefficient matrices, are intimately connected.

• ##### The inverse of a matrix (if it exists) can be found by row reducing the matrix augmented by the identity matrix. math.la.t.mat.inv.augmented
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# Invertible Matrix Theorem

Statements that are equivalent to a square matrix being invertible; examples.

• ##### Definition of matrix inverse math.la.d.mat.inv
Created On
August 21st, 2017
6 years ago
Views
3
Type
Video
Language
English
Content Type
text/html; charset=utf-8

# Matrix inverses

Properties of matrix inversion: inverse of the inverse, inverse of the transpose, inverse of a product; elementary matrices and corresponding row operations; a matrix is invertible if and only if it is row-equivalent to the identity matrix; row-reduction algorithm for computing matrix inverse

• ##### Equivalence theorem: the matrix A row-reduces to the identity matrix. math.la.t.equiv.identity
Created On
August 25th, 2017
6 years ago
Views
3
Type
Video
Language
English
Content Type
text/html; charset=utf-8

# Matrix Inverses and Systems of Linear Equations - A First Course in Linear Algebra

The inverse of a square matrix, and solutions to linear systems with square coefficient matrices, are intimately connected.

• ##### Matrix transpose commutes with matrix inverse. math.la.t.mat.inv.transpose
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# Matrix Inverses and Systems of Linear Equations - A First Course in Linear Algebra

The inverse of a square matrix, and solutions to linear systems with square coefficient matrices, are intimately connected.

• ##### Matrix inverse is an involution. math.la.t.mat.inv.involution
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

# Matrix Inverses and Systems of Linear Equations - A First Course in Linear Algebra

The inverse of a square matrix, and solutions to linear systems with square coefficient matrices, are intimately connected.

• ##### For n-by-n invertible matrices A and B, the product AB is invertible, and (AB)^-1=B^-1 A^-1. math.la.t.mat.inv.shoesandsocks
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# Matrix Inverses and Systems of Linear Equations - A First Course in Linear Algebra

The inverse of a square matrix, and solutions to linear systems with square coefficient matrices, are intimately connected.

• ##### math.la.t.mat.inv.scalar
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

# Finding the inverse of a matrix and using it to solve a linear system

Matrix inverses are motivated as a way to solve a linear system. The general algorithm of finding an inverse by row reducing an augmented matrix is described, and then implemented for a 3x3 matrix. Useful facts about inverses are stated and then illustrated with sample 2x2 matrices. (put first: need Example of finding the inverse of a 3-by-3 matrix by row reducing the augmented matrix)

• ##### The inverse of a matrix can be used to solve a linear system. math.la.t.eqn.mat.inv
Created On
February 19th, 2017
6 years ago
Views
3
Type
Video
Timeframe
Pre-class
Perspective
Introduction
Language
English
Content Type
text/html; charset=utf-8

# Lagrange Interpolation - Ximera

This is a guided discovery of the formula for Lagrange Interpolation, which lets you find the formula for a polynomial which passes through a given set of points.

• ##### Definition of linear transformation, arbitrary vector space math.la.d.lintrans.arb
Created On
June 8th, 2017
6 years ago
Views
2
Type
Handout
Perspective
Application
Language
English
Content Type
text/html; charset=utf-8

# Invertible Matrix Theorem

Statements that are equivalent to a square matrix being invertible; examples.

• ##### Definition of matrix inverse math.la.d.mat.inv
Created On
August 21st, 2017
6 years ago
Views
3
Type
Video
Language
English
Content Type
text/html; charset=utf-8

# Matrix Inverses, definition and examples

Definition of the inverse of a matrix, examples, uniqueness; formula for the inverse of a 2x2 matrix; determinant of a 2x2 matrix; using the inverse to solve a system of linear equations.

• ##### The inverse of a matrix can be used to solve a linear system. math.la.t.eqn.mat.inv
Created On
August 22nd, 2017
6 years ago
Views
4
Type
Video
Language
English
Content Type
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# Matrix Inverses and Nonsingular Matrices - A First Course in Linear Algebra

We saw in Theorem CINM that if a square matrix $$A$$ is nonsingular, then there is a matrix $$B$$ so that $$AB=I_n\text{.}$$ In other words, $$B$$ is halfway to being an inverse of $$A\text{.}$$ We will see in this section that $$B$$ automatically fulfills the second condition ($$BA=I_n$$). Example MWIAA showed us that the coefficient matrix from Archetype A had no inverse. Not coincidentally, this coefficient matrix is singular. We will make all these connections precise now. Not many examples or definitions in this section, just theorems.

• ##### math.la.t.mat.prod.nonsingular
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# Matrix Inverses and Nonsingular Matrices - A First Course in Linear Algebra

We saw in Theorem CINM that if a square matrix $$A$$ is nonsingular, then there is a matrix $$B$$ so that $$AB=I_n\text{.}$$ In other words, $$B$$ is halfway to being an inverse of $$A\text{.}$$ We will see in this section that $$B$$ automatically fulfills the second condition ($$BA=I_n$$). Example MWIAA showed us that the coefficient matrix from Archetype A had no inverse. Not coincidentally, this coefficient matrix is singular. We will make all these connections precise now. Not many examples or definitions in this section, just theorems.

• ##### The inverse of a matrix can be used to solve a linear system. math.la.t.eqn.mat.inv
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# Matrix Inverses and Nonsingular Matrices - A First Course in Linear Algebra

We saw in Theorem CINM that if a square matrix $$A$$ is nonsingular, then there is a matrix $$B$$ so that $$AB=I_n\text{.}$$ In other words, $$B$$ is halfway to being an inverse of $$A\text{.}$$ We will see in this section that $$B$$ automatically fulfills the second condition ($$BA=I_n$$). Example MWIAA showed us that the coefficient matrix from Archetype A had no inverse. Not coincidentally, this coefficient matrix is singular. We will make all these connections precise now. Not many examples or definitions in this section, just theorems.

• ##### Equivalence theorem: the matrix A has an inverse. math.la.t.equiv.inv
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# Matrix Inverses and Nonsingular Matrices - A First Course in Linear Algebra

We saw in Theorem CINM that if a square matrix $$A$$ is nonsingular, then there is a matrix $$B$$ so that $$AB=I_n\text{.}$$ In other words, $$B$$ is halfway to being an inverse of $$A\text{.}$$ We will see in this section that $$B$$ automatically fulfills the second condition ($$BA=I_n$$). Example MWIAA showed us that the coefficient matrix from Archetype A had no inverse. Not coincidentally, this coefficient matrix is singular. We will make all these connections precise now. Not many examples or definitions in this section, just theorems.

• ##### math.la.t.mat.inv.oneside
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# Matrix Inverses and Nonsingular Matrices - A First Course in Linear Algebra

We saw in Theorem CINM that if a square matrix $$A$$ is nonsingular, then there is a matrix $$B$$ so that $$AB=I_n\text{.}$$ In other words, $$B$$ is halfway to being an inverse of $$A\text{.}$$ We will see in this section that $$B$$ automatically fulfills the second condition ($$BA=I_n$$). Example MWIAA showed us that the coefficient matrix from Archetype A had no inverse. Not coincidentally, this coefficient matrix is singular. We will make all these connections precise now. Not many examples or definitions in this section, just theorems.

• ##### math.la.t.mat.unitary.col.orthogonal
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6 years ago
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# Matrix Inverses and Nonsingular Matrices - A First Course in Linear Algebra

We saw in Theorem CINM that if a square matrix $$A$$ is nonsingular, then there is a matrix $$B$$ so that $$AB=I_n\text{.}$$ In other words, $$B$$ is halfway to being an inverse of $$A\text{.}$$ We will see in this section that $$B$$ automatically fulfills the second condition ($$BA=I_n$$). Example MWIAA showed us that the coefficient matrix from Archetype A had no inverse. Not coincidentally, this coefficient matrix is singular. We will make all these connections precise now. Not many examples or definitions in this section, just theorems.

• ##### math.la.t.mat.unitary.innerproduct
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6 years ago
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# Matrix Inverses and Nonsingular Matrices - A First Course in Linear Algebra

We saw in Theorem CINM that if a square matrix $$A$$ is nonsingular, then there is a matrix $$B$$ so that $$AB=I_n\text{.}$$ In other words, $$B$$ is halfway to being an inverse of $$A\text{.}$$ We will see in this section that $$B$$ automatically fulfills the second condition ($$BA=I_n$$). Example MWIAA showed us that the coefficient matrix from Archetype A had no inverse. Not coincidentally, this coefficient matrix is singular. We will make all these connections precise now. Not many examples or definitions in this section, just theorems.

• ##### math.la.t.mat.unitary.inv
GFDL-1.2
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September 11th, 2017
6 years ago
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# Matrix Inverses and Nonsingular Matrices - A First Course in Linear Algebra

We saw in Theorem CINM that if a square matrix $$A$$ is nonsingular, then there is a matrix $$B$$ so that $$AB=I_n\text{.}$$ In other words, $$B$$ is halfway to being an inverse of $$A\text{.}$$ We will see in this section that $$B$$ automatically fulfills the second condition ($$BA=I_n$$). Example MWIAA showed us that the coefficient matrix from Archetype A had no inverse. Not coincidentally, this coefficient matrix is singular. We will make all these connections precise now. Not many examples or definitions in this section, just theorems.

• ##### Definition of unitary matrix math.la.d.mat.unitary
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# Matrix equations as motivation for basis and span

Advice to instructors for in-class activities on matrix-vector multiplication and translating between the various equivalent notation forms of linear systems, and suggestions for how this topic can be used to motivate future topics.

• ##### The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A. math.la.t.mat.eqn.lincomb
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# Matrix equations with zero or infinitely many solutions

A 3x3 matrix equation Ax=b is solved for two different values of b. In one case there is no solution, and in another there are infinitely many solutions. These examples illustrate a theorem about linear combinations of the columns of the matrix A.

• ##### Definition of matrix equation math.la.d.mat.eqn
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# Linear combinations of vectors

The linear combination of a set of vectors is defined. Determine if a vector in R^2 is in the span of two other vectors. The span of a set of vectors is related to the columns of a matrix. (need topic: Determine if a vector in R^2 is in the span of two other vectors.)

• ##### The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A. math.la.t.mat.eqn.lincomb
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# Matrix equations and spanning sets

Equivalent statements for a matrix A: for every right-hand side b, the system Ax=b has a solution; every b is a linear combination of the columns of A; the span of the columns of A is maximal; A has a pivot position in every row.

• ##### The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A. math.la.t.mat.eqn.lincomb
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# Column space of a matrix

Definition of the column space of a matrix; column space is a subspace; comparison to the null space; definition of a linear transformation between vector spaces; definition of kernel and range of a linear transformation

• ##### Definition of range of linear transformation, arbitrary vector space math.la.d.lintrans.range.arb
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# Linear Combinations - A First Course in Linear Algebra

In Section VO we defined vector addition and scalar multiplication. These two operations combine nicely to give us a construction known as a linear combination, a construct that we will work with throughout this course.

• ##### The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A. math.la.t.mat.eqn.lincomb
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# Column and Row Spaces - A First Course in Linear Algebra

A matrix-vector product (Definition MVP) is a linear combination of the columns of the matrix and this allows us to connect matrix multiplication with systems of equations via Theorem SLSLC. Row operations are linear combinations of the rows of a matrix, and of course, reduced row-echelon form (Definition RREF) is also intimately related to solving systems of equations. In this section we will formalize these ideas with two key definitions of sets of vectors derived from a matrix.

• ##### The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A. math.la.t.mat.eqn.lincomb
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# Column and Row Spaces - A First Course in Linear Algebra

A matrix-vector product (Definition MVP) is a linear combination of the columns of the matrix and this allows us to connect matrix multiplication with systems of equations via Theorem SLSLC. Row operations are linear combinations of the rows of a matrix, and of course, reduced row-echelon form (Definition RREF) is also intimately related to solving systems of equations. In this section we will formalize these ideas with two key definitions of sets of vectors derived from a matrix.

• ##### Definition of column space of a matrix math.la.d.mat.col_space
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# Dimension of vector spaces

Basis theorem: for an n-dimensional vector space any linearly independent set with n elements is a basis, as is any spanning set with n elements; dimension of the column space of a matrix equals the number of pivot columns of the matrix; dimension of the null space of a matrix equals the number of free variables of the matrix

• ##### The pivot columns of a matrix are a basis for the column space. math.la.t.mat.col_space.pivot
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# Bases for the nullspace and column space of a matrix

The pivot columns of a matrix form a basis for its column space; nullspace of a matrix equals the nullspace of its reduced row-echelon form.

• ##### The pivot columns of a matrix are a basis for the column space. math.la.t.mat.col_space.pivot
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# Column and Row Spaces - A First Course in Linear Algebra

A matrix-vector product (Definition MVP) is a linear combination of the columns of the matrix and this allows us to connect matrix multiplication with systems of equations via Theorem SLSLC. Row operations are linear combinations of the rows of a matrix, and of course, reduced row-echelon form (Definition RREF) is also intimately related to solving systems of equations. In this section we will formalize these ideas with two key definitions of sets of vectors derived from a matrix.

• ##### The pivot columns of a matrix are a basis for the column space. math.la.t.mat.col_space.pivot
GFDL-1.2
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# Matrix equations and spanning sets

Equivalent statements for a matrix A: for every right-hand side b, the system Ax=b has a solution; every b is a linear combination of the columns of A; the span of the columns of A is maximal; A has a pivot position in every row.

• ##### The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A. math.la.t.mat.eqn.lincomb
Created On
August 25th, 2017
6 years ago
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2
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# Column and Row Spaces - A First Course in Linear Algebra

A matrix-vector product (Definition MVP) is a linear combination of the columns of the matrix and this allows us to connect matrix multiplication with systems of equations via Theorem SLSLC. Row operations are linear combinations of the rows of a matrix, and of course, reduced row-echelon form (Definition RREF) is also intimately related to solving systems of equations. In this section we will formalize these ideas with two key definitions of sets of vectors derived from a matrix.

• ##### Equivalence theorem: the columns of A span R^n (or C^n). math.la.t.equiv.col.span
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6 years ago
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# Column and Row Spaces - A First Course in Linear Algebra

A matrix-vector product (Definition MVP) is a linear combination of the columns of the matrix and this allows us to connect matrix multiplication with systems of equations via Theorem SLSLC. Row operations are linear combinations of the rows of a matrix, and of course, reduced row-echelon form (Definition RREF) is also intimately related to solving systems of equations. In this section we will formalize these ideas with two key definitions of sets of vectors derived from a matrix.

• ##### math.la.t.mat.row_space.pivot
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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Textbook
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English
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# Column and Row Spaces - A First Course in Linear Algebra

A matrix-vector product (Definition MVP) is a linear combination of the columns of the matrix and this allows us to connect matrix multiplication with systems of equations via Theorem SLSLC. Row operations are linear combinations of the rows of a matrix, and of course, reduced row-echelon form (Definition RREF) is also intimately related to solving systems of equations. In this section we will formalize these ideas with two key definitions of sets of vectors derived from a matrix.

• ##### Row equivalent matrices have the same row space. math.la.d.mat.row_space.row_equiv
GFDL-1.2
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# Column and Row Spaces - A First Course in Linear Algebra

A matrix-vector product (Definition MVP) is a linear combination of the columns of the matrix and this allows us to connect matrix multiplication with systems of equations via Theorem SLSLC. Row operations are linear combinations of the rows of a matrix, and of course, reduced row-echelon form (Definition RREF) is also intimately related to solving systems of equations. In this section we will formalize these ideas with two key definitions of sets of vectors derived from a matrix.

• ##### Definition of row space of a matrix math.la.d.mat.row_space
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# Four Subsets - A First Course in Linear Algebra

There are four natural subsets associated with a matrix. We have met three already: the null space, the column space and the row space. In this section we will introduce a fourth, the left null space. The objective of this section is to describe one procedure that will allow us to find linearly independent sets that span each of these four sets of column vectors. Along the way, we will make a connection with the inverse of a matrix, so Theorem FS will tie together most all of this chapter (and the entire course so far).

• ##### Definition of matrix null space (left) math.la.d.mat.null_space.left
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# Four Subsets - A First Course in Linear Algebra

There are four natural subsets associated with a matrix. We have met three already: the null space, the column space and the row space. In this section we will introduce a fourth, the left null space. The objective of this section is to describe one procedure that will allow us to find linearly independent sets that span each of these four sets of column vectors. Along the way, we will make a connection with the inverse of a matrix, so Theorem FS will tie together most all of this chapter (and the entire course so far).

• ##### math.la.t.mat.erref.of
GFDL-1.2
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6 years ago
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# Four Subsets - A First Course in Linear Algebra

There are four natural subsets associated with a matrix. We have met three already: the null space, the column space and the row space. In this section we will introduce a fourth, the left null space. The objective of this section is to describe one procedure that will allow us to find linearly independent sets that span each of these four sets of column vectors. Along the way, we will make a connection with the inverse of a matrix, so Theorem FS will tie together most all of this chapter (and the entire course so far).

• ##### math.la.d.mat.erref.of
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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# Four Subsets - A First Course in Linear Algebra

There are four natural subsets associated with a matrix. We have met three already: the null space, the column space and the row space. In this section we will introduce a fourth, the left null space. The objective of this section is to describe one procedure that will allow us to find linearly independent sets that span each of these four sets of column vectors. Along the way, we will make a connection with the inverse of a matrix, so Theorem FS will tie together most all of this chapter (and the entire course so far).

• ##### math.la.c.mat.col_space.row_reduce
GFDL-1.2
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September 11th, 2017
6 years ago
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# Four Subsets - A First Course in Linear Algebra

There are four natural subsets associated with a matrix. We have met three already: the null space, the column space and the row space. In this section we will introduce a fourth, the left null space. The objective of this section is to describe one procedure that will allow us to find linearly independent sets that span each of these four sets of column vectors. Along the way, we will make a connection with the inverse of a matrix, so Theorem FS will tie together most all of this chapter (and the entire course so far).

• ##### math.la.t.mat.erref.spaces
GFDL-1.2
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# Vector space Part 1: what makes R^n a "space"

This video kicks off the series of videos on vector spaces. We begin by summarizing the essential properties of R^n.

• ##### Axioms of a vector space, arbitrary vector space math.la.d.vsp.axioms.arb
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# Vector spaces Part 2: What makes R^n a "space" continued

In this video we continue to list the properties of R^n. The 10 properties listed in this video and the previous video will be used to define a general vecto...

• ##### Axioms of a vector space, arbitrary vector space math.la.d.vsp.axioms.arb
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6 years ago
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# Vector space Part 3: definition of a vector space

The concept of a vector space is somewhat abstract, and under this definition, a lot of objects such as polynomials, functions, etc., can be considered as vectors. This video explains the definition of a general vector space. In later videos we will look at more examples.

• ##### Axioms of a vector space, arbitrary vector space math.la.d.vsp.axioms.arb
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# Vector spaces

Definition of a (real) vector space; properties of the zero vector and the additive inverse in relation to scalar multiplication

• ##### The negative of a vector equals the vector multiplied by -1. math.la.t.vsp.vector.negative
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# Matrix Operations - A First Course in Linear Algebra

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

• ##### math.la.e.vsp.mat.m_by_n
GFDL-1.2
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# Vector Spaces - A First Course in Linear Algebra

In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

• ##### math.la.e.vsp.polynomial.leq_n
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# Vector Spaces - A First Course in Linear Algebra

In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

• ##### Definition of vector, arbitrary vector space math.la.d.vec.arb
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# Vector Spaces - A First Course in Linear Algebra

In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

• ##### math.la.e.vsp.cn
GFDL-1.2
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September 11th, 2017
6 years ago
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# Vector Spaces - A First Course in Linear Algebra

In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

• ##### math.la.e.vsp.function
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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# Vector Spaces - A First Course in Linear Algebra

In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

• ##### math.la.e.vsp.sequence
GFDL-1.2
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6 years ago
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# Vector Spaces - A First Course in Linear Algebra

In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

• ##### math.la.e.vsp.crazy
GFDL-1.2
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# Vector Spaces - A First Course in Linear Algebra

In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

• ##### math.la.e.vsp.z
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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# Vector Spaces - A First Course in Linear Algebra

In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

• ##### math.la.e.vsp.mat.m_by_n
GFDL-1.2
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September 11th, 2017
6 years ago
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# Vector spaces

Definition of a (real) vector space; properties of the zero vector and the additive inverse in relation to scalar multiplication

• ##### The negative of a vector equals the vector multiplied by -1. math.la.t.vsp.vector.negative
Created On
August 25th, 2017
6 years ago
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# Vector Spaces - A First Course in Linear Algebra

In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

• ##### The negative of a vector equals the vector multiplied by -1. math.la.t.vsp.vector.negative
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# Vector Spaces - A First Course in Linear Algebra

In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

• ##### The zero scalar multiplied by any vector equals the zero vector. math.la.t.vsp.scalar.mult.z
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# Vector Spaces - A First Course in Linear Algebra

In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

• ##### math.la.t.vsp.z.unique
GFDL-1.2
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# Vector Spaces - A First Course in Linear Algebra

In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

• ##### math.la.d.vsp.vector.negative.unique
GFDL-1.2
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September 11th, 2017
6 years ago
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# Vector Spaces - A First Course in Linear Algebra

In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

• ##### The zero vector multiplied by any scalar equals the zero vector. math.la.t.vsp.vector.mult.z
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# Vector Spaces - A First Course in Linear Algebra

In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

• ##### math.la.t.vsp.mult.z
GFDL-1.2
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September 11th, 2017
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# Subspaces Part 1: definition of a subspace

Preliminaries: 1. What is a subset? 2. How to verify a set is a subset of another set? 3. Notations and language of set theory related to subsets. In this video, we introduce the definition of a subspace. We go through a preliminary example to figure out what do subspaces of R^2 look like, and we will continue to talk about how to verify a subset of a vector space is a subspace in later videos.

• ##### Definition of subspace, arbitrary vector space math.la.d.vsp.subspace.arb
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# Testing if a subset of a vector space is a subspace, the 2 axioms involved and why

In this video, I'll explain why we only need to test 2 axioms (among the 10 axioms in the definition of a vector space) when figuring out if a subset is a subspace.

• ##### Definition of subspace, arbitrary vector space math.la.d.vsp.subspace.arb
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# Vector subspaces

Definition of a subspace of a vector space; examples; span of vectors is a subspace.

• ##### Definition of zero subspace, arbitrary vector space math.la.d.vsp.subspace.z
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# Subspaces - A First Course in Linear Algebra

A subspace is a vector space that is contained within another vector space. So every subspace is a vector space in its own right, but it is also defined relative to some other (larger) vector space. We will discover shortly that we are already familiar with a wide variety of subspaces from previous sections.

• ##### Definition of subspace, arbitrary vector space math.la.d.vsp.subspace.arb
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# Subspaces - A First Course in Linear Algebra

A subspace is a vector space that is contained within another vector space. So every subspace is a vector space in its own right, but it is also defined relative to some other (larger) vector space. We will discover shortly that we are already familiar with a wide variety of subspaces from previous sections.

• ##### math.la.d.vsp.subspace.z.arb
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# Subspaces - A First Course in Linear Algebra

A subspace is a vector space that is contained within another vector space. So every subspace is a vector space in its own right, but it is also defined relative to some other (larger) vector space. We will discover shortly that we are already familiar with a wide variety of subspaces from previous sections.

• ##### The null space of a matrix is a subspace of R^n (or C^n). math.la.t.mat.null_space.rncn
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# Subspaces - A First Course in Linear Algebra

A subspace is a vector space that is contained within another vector space. So every subspace is a vector space in its own right, but it is also defined relative to some other (larger) vector space. We will discover shortly that we are already familiar with a wide variety of subspaces from previous sections.

• ##### math.la.t.vsp.subspace.lincomb.arb
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# Vectors and their linear combinations in R^n

Definition of a vector; vector addition; scalar multiplication; visualization in R^2 and R^3; vector space axioms; linear combinations; span.

• ##### Example of writing a given vector in R^3 as a linear combination of given vectors math.la.e.vec.lincomb.weight.solve.r3
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# Subspaces - A First Course in Linear Algebra

A subspace is a vector space that is contained within another vector space. So every subspace is a vector space in its own right, but it is also defined relative to some other (larger) vector space. We will discover shortly that we are already familiar with a wide variety of subspaces from previous sections.

• ##### math.la.t.vec.span.subspace.arb
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# Subspaces - A First Course in Linear Algebra

A subspace is a vector space that is contained within another vector space. So every subspace is a vector space in its own right, but it is also defined relative to some other (larger) vector space. We will discover shortly that we are already familiar with a wide variety of subspaces from previous sections.

• ##### Definition of span of a set of vectors, arbitrary vector space math.la.d.vec.span.arb
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# Subspaces - A First Course in Linear Algebra

A subspace is a vector space that is contained within another vector space. So every subspace is a vector space in its own right, but it is also defined relative to some other (larger) vector space. We will discover shortly that we are already familiar with a wide variety of subspaces from previous sections.

• ##### Definition of linear combination of vectors, arbitrary vector space math.la.d.vec.lincomb.arb
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# Subspaces - A First Course in Linear Algebra

A subspace is a vector space that is contained within another vector space. So every subspace is a vector space in its own right, but it is also defined relative to some other (larger) vector space. We will discover shortly that we are already familiar with a wide variety of subspaces from previous sections.

• ##### math.la.e.vsp.row
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# Subspaces - A First Course in Linear Algebra

A subspace is a vector space that is contained within another vector space. So every subspace is a vector space in its own right, but it is also defined relative to some other (larger) vector space. We will discover shortly that we are already familiar with a wide variety of subspaces from previous sections.

• ##### math.la.e.vsp.col
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# Subspaces - A First Course in Linear Algebra

A subspace is a vector space that is contained within another vector space. So every subspace is a vector space in its own right, but it is also defined relative to some other (larger) vector space. We will discover shortly that we are already familiar with a wide variety of subspaces from previous sections.

• ##### math.la.t.mat.null_space.left.rncn
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# Linear Independence and Spanning Sets - A First Course in Linear Algebra

A vector space is defined as a set with two operations, meeting ten properties (Definition VS). Just as the definition of span of a set of vectors only required knowing how to add vectors and how to multiply vectors by scalars, so it is with linear independence. A definition of a linearly independent set of vectors in an arbitrary vector space only requires knowing how to form linear combinations and equating these with the zero vector. Since every vector space must have a zero vector (Property Z), we always have a zero vector at our disposal.

• ##### Definition of linearly indepentent set of vectors: if a linear combination is zero, then every coefficient is zero, arbitrary vector space. math.la.d.vec.linindep.arb
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# Linear Independence and Spanning Sets - A First Course in Linear Algebra

A vector space is defined as a set with two operations, meeting ten properties (Definition VS). Just as the definition of span of a set of vectors only required knowing how to add vectors and how to multiply vectors by scalars, so it is with linear independence. A definition of a linearly independent set of vectors in an arbitrary vector space only requires knowing how to form linear combinations and equating these with the zero vector. Since every vector space must have a zero vector (Property Z), we always have a zero vector at our disposal.

• ##### math.la.d.vec.lindep.relation.trvial.rep
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# Linear Independence and Spanning Sets - A First Course in Linear Algebra

A vector space is defined as a set with two operations, meeting ten properties (Definition VS). Just as the definition of span of a set of vectors only required knowing how to add vectors and how to multiply vectors by scalars, so it is with linear independence. A definition of a linearly independent set of vectors in an arbitrary vector space only requires knowing how to form linear combinations and equating these with the zero vector. Since every vector space must have a zero vector (Property Z), we always have a zero vector at our disposal.

• ##### Definition of spanning set for a subspace, arbitrary vector space math.la.d.vsp.span.set.arb
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# Coordinate systems

Representation (unique) of a vector in terms of a basis for a vector space yields coordinates relative to the basis; change of basis and corresponding change of coordinate matrix

• ##### Definition of coordinates relative to a given basis, coordinate vector space math.la.d.vsp.basis.relative.coord
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# Linear Independence and Spanning Sets - A First Course in Linear Algebra

A vector space is defined as a set with two operations, meeting ten properties (Definition VS). Just as the definition of span of a set of vectors only required knowing how to add vectors and how to multiply vectors by scalars, so it is with linear independence. A definition of a linearly independent set of vectors in an arbitrary vector space only requires knowing how to form linear combinations and equating these with the zero vector. Since every vector space must have a zero vector (Property Z), we always have a zero vector at our disposal.

• ##### Each vector can be written uniquely as a linear combination of vectors from a given basis. math.la.t.vsp.basis.coord.unique
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# Bases - A First Course in Linear Algebra

A basis of a vector space is one of the most useful concepts in linear algebra. It often provides a concise, finite description of an infinite vector space.

• ##### Definition of the standard basis of the polynomials of degree at most n math.la.d.vsp.basis.standard.leq_n
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# Bases - A First Course in Linear Algebra

A basis of a vector space is one of the most useful concepts in linear algebra. It often provides a concise, finite description of an infinite vector space.

• ##### Definition of basis of a vector space (or subspace), arbitrary vector space math.la.d.vsp.basis.arb
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# Bases - A First Course in Linear Algebra

A basis of a vector space is one of the most useful concepts in linear algebra. It often provides a concise, finite description of an infinite vector space.

• ##### math.la.t.vsp.basis.standard.rncn
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# Bases - A First Course in Linear Algebra

A basis of a vector space is one of the most useful concepts in linear algebra. It often provides a concise, finite description of an infinite vector space.

• ##### math.la.d.vsp.basis.standard.m_by_n
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# Bases - A First Course in Linear Algebra

A basis of a vector space is one of the most useful concepts in linear algebra. It often provides a concise, finite description of an infinite vector space.

• ##### Equivalence theorem: the columns of A are a basis for R^n (or C^n). math.la.t.equiv.col.basis
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# Bases - A First Course in Linear Algebra

A basis of a vector space is one of the most useful concepts in linear algebra. It often provides a concise, finite description of an infinite vector space.

• ##### Formula for the coordinates of a vector with respect to an orthogonal basis. math.la.t.subspace.basis.orthogonal
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# Bases - A First Course in Linear Algebra

A basis of a vector space is one of the most useful concepts in linear algebra. It often provides a concise, finite description of an infinite vector space.

• ##### math.la.t.mat.unitary.basis.orthogonal
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# Dimension - A First Course in Linear Algebra

Almost every vector space we have encountered has been infinite in size (an exception is Example VSS). But some are bigger and richer than others. Dimension, once suitably defined, will be a measure of the size of a vector space, and a useful tool for studying its properties. You probably already have a rough notion of what a mathematical definition of dimension might be — try to forget these imprecise ideas and go with the new ones given here.

• ##### Definition of dimension of a vector space (or subspace) being finite or infinite, arbitrary vector space math.la.d.vsp.dim.finite_infinite.arb
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# Properties of Dimension - A First Course in Linear Algebra

Once the dimension of a vector space is known, then the determination of whether or not a set of vectors is linearly independent, or if it spans the vector space, can often be much easier. In this section we will state a workhorse theorem and then apply it to the column space and row space of a matrix. It will also help us describe a super-basis for $$\complex{m}\text{.}$$

• ##### math.la.t.vsp.dim.span.linindep.arb
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# Dimension - A First Course in Linear Algebra

Almost every vector space we have encountered has been infinite in size (an exception is Example VSS). But some are bigger and richer than others. Dimension, once suitably defined, will be a measure of the size of a vector space, and a useful tool for studying its properties. You probably already have a rough notion of what a mathematical definition of dimension might be — try to forget these imprecise ideas and go with the new ones given here.

• ##### A set of vectors containing more elements than the dimension of the space must be linearly dependent, arbitrary vector space. math.la.t.vsp.dim.more.lindep.arb
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# Dimension - A First Course in Linear Algebra

Almost every vector space we have encountered has been infinite in size (an exception is Example VSS). But some are bigger and richer than others. Dimension, once suitably defined, will be a measure of the size of a vector space, and a useful tool for studying its properties. You probably already have a rough notion of what a mathematical definition of dimension might be — try to forget these imprecise ideas and go with the new ones given here.

• ##### Every basis for a vector space contains the same number of elements, arbitrary vector space. math.la.t.vsp.dim.arb
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# In-class activity: Rank nullity theorem

Students answer multiple questions on the rank and dimension of the null space in a variety of situations to discover the connection between these dimensions leading to the Rank-Nullity Theorem.

• ##### If A is a matrix, then the rank of A plus the nullity of A equals the number of columns of A. math.la.t.mat.ranknullity
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# Dimension - A First Course in Linear Algebra

Almost every vector space we have encountered has been infinite in size (an exception is Example VSS). But some are bigger and richer than others. Dimension, once suitably defined, will be a measure of the size of a vector space, and a useful tool for studying its properties. You probably already have a rough notion of what a mathematical definition of dimension might be — try to forget these imprecise ideas and go with the new ones given here.

• ##### Definition of nullity of a matrix math.la.d.mat.nullity
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# Dimension - A First Course in Linear Algebra

Almost every vector space we have encountered has been infinite in size (an exception is Example VSS). But some are bigger and richer than others. Dimension, once suitably defined, will be a measure of the size of a vector space, and a useful tool for studying its properties. You probably already have a rough notion of what a mathematical definition of dimension might be — try to forget these imprecise ideas and go with the new ones given here.

• ##### math.la.t.mat.rank.pivot
GFDL-1.2
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6 years ago
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# Dimension - A First Course in Linear Algebra

Almost every vector space we have encountered has been infinite in size (an exception is Example VSS). But some are bigger and richer than others. Dimension, once suitably defined, will be a measure of the size of a vector space, and a useful tool for studying its properties. You probably already have a rough notion of what a mathematical definition of dimension might be — try to forget these imprecise ideas and go with the new ones given here.

• ##### Definition of rank of a matrix math.la.d.mat.rank
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# Dimension - A First Course in Linear Algebra

Almost every vector space we have encountered has been infinite in size (an exception is Example VSS). But some are bigger and richer than others. Dimension, once suitably defined, will be a measure of the size of a vector space, and a useful tool for studying its properties. You probably already have a rough notion of what a mathematical definition of dimension might be — try to forget these imprecise ideas and go with the new ones given here.

• ##### If A is a matrix, then the rank of A plus the nullity of A equals the number of columns of A. math.la.t.mat.ranknullity
GFDL-1.2
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September 11th, 2017
6 years ago
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# Dimension - A First Course in Linear Algebra

Almost every vector space we have encountered has been infinite in size (an exception is Example VSS). But some are bigger and richer than others. Dimension, once suitably defined, will be a measure of the size of a vector space, and a useful tool for studying its properties. You probably already have a rough notion of what a mathematical definition of dimension might be — try to forget these imprecise ideas and go with the new ones given here.

• ##### Equivalence theorem: the nullity of the matrix A is zero. math.la.t.equiv.nullity
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# Dimension of vector spaces

Basis theorem: for an n-dimensional vector space any linearly independent set with n elements is a basis, as is any spanning set with n elements; dimension of the column space of a matrix equals the number of pivot columns of the matrix; dimension of the null space of a matrix equals the number of free variables of the matrix

• ##### The pivot columns of a matrix are a basis for the column space. math.la.t.mat.col_space.pivot
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# Properties of Dimension - A First Course in Linear Algebra

Once the dimension of a vector space is known, then the determination of whether or not a set of vectors is linearly independent, or if it spans the vector space, can often be much easier. In this section we will state a workhorse theorem and then apply it to the column space and row space of a matrix. It will also help us describe a super-basis for $$\complex{m}\text{.}$$

• ##### math.la.t.vsp.linindep.extend
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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# Properties of Dimension - A First Course in Linear Algebra

Once the dimension of a vector space is known, then the determination of whether or not a set of vectors is linearly independent, or if it spans the vector space, can often be much easier. In this section we will state a workhorse theorem and then apply it to the column space and row space of a matrix. It will also help us describe a super-basis for $$\complex{m}\text{.}$$

• ##### The dimension of a subspace is less than or equal to the dimension of the whole space, arbitrary vector space. math.la.t.vsp.subspace.dim.arb
GFDL-1.2
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6 years ago
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# Properties of Dimension - A First Course in Linear Algebra

Once the dimension of a vector space is known, then the determination of whether or not a set of vectors is linearly independent, or if it spans the vector space, can often be much easier. In this section we will state a workhorse theorem and then apply it to the column space and row space of a matrix. It will also help us describe a super-basis for $$\complex{m}\text{.}$$

• ##### math.la.t.vsp.dim.span.linindep.arb
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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# Properties of Dimension - A First Course in Linear Algebra

Once the dimension of a vector space is known, then the determination of whether or not a set of vectors is linearly independent, or if it spans the vector space, can often be much easier. In this section we will state a workhorse theorem and then apply it to the column space and row space of a matrix. It will also help us describe a super-basis for $$\complex{m}\text{.}$$

• ##### math.la.t.vsp.subspace.dim.equal
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
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English
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# Dimension - A First Course in Linear Algebra

Almost every vector space we have encountered has been infinite in size (an exception is Example VSS). But some are bigger and richer than others. Dimension, once suitably defined, will be a measure of the size of a vector space, and a useful tool for studying its properties. You probably already have a rough notion of what a mathematical definition of dimension might be — try to forget these imprecise ideas and go with the new ones given here.

• ##### A set of vectors containing more elements than the dimension of the space must be linearly dependent, arbitrary vector space. math.la.t.vsp.dim.more.lindep.arb
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
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# Properties of Dimension - A First Course in Linear Algebra

Once the dimension of a vector space is known, then the determination of whether or not a set of vectors is linearly independent, or if it spans the vector space, can often be much easier. In this section we will state a workhorse theorem and then apply it to the column space and row space of a matrix. It will also help us describe a super-basis for $$\complex{m}\text{.}$$

• ##### The row space and the column space of a matrix have the same dimension. math.la.t.mat.row_space.col_space
GFDL-1.2
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September 11th, 2017
6 years ago
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# Properties of Dimension - A First Course in Linear Algebra

Once the dimension of a vector space is known, then the determination of whether or not a set of vectors is linearly independent, or if it spans the vector space, can often be much easier. In this section we will state a workhorse theorem and then apply it to the column space and row space of a matrix. It will also help us describe a super-basis for $$\complex{m}\text{.}$$

• ##### math.la.t.mat.erref.dimension
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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# Matrix inverses

Properties of matrix inversion: inverse of the inverse, inverse of the transpose, inverse of a product; elementary matrices and corresponding row operations; a matrix is invertible if and only if it is row-equivalent to the identity matrix; row-reduction algorithm for computing matrix inverse

• ##### Equivalence theorem: the matrix A row-reduces to the identity matrix. math.la.t.equiv.identity
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# Determinant of a Matrix - A First Course in Linear Algebra

Before we define the determinant of a matrix, we take a slight detour to introduce elementary matrices. These will bring us back to the beginning of the course and our old friend, row operations.

• ##### math.la.d.mat.elementary.prod
GFDL-1.2
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# Determinant of a Matrix - A First Course in Linear Algebra

Before we define the determinant of a matrix, we take a slight detour to introduce elementary matrices. These will bring us back to the beginning of the course and our old friend, row operations.

• ##### math.la.t.mat.mult.elementary
GFDL-1.2
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# Determinant of a Matrix - A First Course in Linear Algebra

Before we define the determinant of a matrix, we take a slight detour to introduce elementary matrices. These will bring us back to the beginning of the course and our old friend, row operations.

• ##### Elementary matrices are invertible. math.la.d.mat.elementary.inv
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# Determinant of a Matrix - A First Course in Linear Algebra

Before we define the determinant of a matrix, we take a slight detour to introduce elementary matrices. These will bring us back to the beginning of the course and our old friend, row operations.

• ##### Definition of elementary matrix math.la.d.mat.elementary
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# The inverse of 2x2 matrices

The formula for the inverse of a 2x2 matrix is derived. (need tag for that formula)

• ##### Formula for the determinant of a 2-by-2 matrix. math.la.t.mat.det.2x2
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February 17th, 2017
6 years ago
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# Matrix Inverses, definition and examples

Definition of the inverse of a matrix, examples, uniqueness; formula for the inverse of a 2x2 matrix; determinant of a 2x2 matrix; using the inverse to solve a system of linear equations.

• ##### The inverse of a matrix can be used to solve a linear system. math.la.t.eqn.mat.inv
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August 22nd, 2017
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# Determinant of a Matrix - A First Course in Linear Algebra

Before we define the determinant of a matrix, we take a slight detour to introduce elementary matrices. These will bring us back to the beginning of the course and our old friend, row operations.

• ##### Definition of determinant of a matrix as a cofactor expansion across the first row math.la.d.mat.det.cofactor
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# Determinant of a Matrix - A First Course in Linear Algebra

Before we define the determinant of a matrix, we take a slight detour to introduce elementary matrices. These will bring us back to the beginning of the course and our old friend, row operations.

• ##### Definition of cofactor of a matrix math.la.d.mat.cofactor
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# Determinant of a Matrix - A First Course in Linear Algebra

Before we define the determinant of a matrix, we take a slight detour to introduce elementary matrices. These will bring us back to the beginning of the course and our old friend, row operations.

• ##### math.la.c.mat.det
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6 years ago
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# Determinant of a Matrix - A First Course in Linear Algebra

Before we define the determinant of a matrix, we take a slight detour to introduce elementary matrices. These will bring us back to the beginning of the course and our old friend, row operations.

• ##### Formula for the determinant of a 2-by-2 matrix. math.la.t.mat.det.2x2
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# Determinants and their relation to column operations and products

Determinant of the transpose equals the determinant of the original matrix; rescaling a column rescales the determinant by the same factor; interchanging two columns changes the sign of the determinant; adding multiple of one column to another leaves determinant unchanged; determinant of the product of two matrices equals product of the two determinants

• ##### Theorem describing the effect of elementary row operations on the determinant of a matrix. math.la.t.mat.det.elementaryoperations
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# Determinant of a Matrix - A First Course in Linear Algebra

Before we define the determinant of a matrix, we take a slight detour to introduce elementary matrices. These will bring us back to the beginning of the course and our old friend, row operations.

• ##### math.la.t.mat.det.cofactor.row
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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# Determinant of a Matrix - A First Course in Linear Algebra

Before we define the determinant of a matrix, we take a slight detour to introduce elementary matrices. These will bring us back to the beginning of the course and our old friend, row operations.

• ##### math.la.t.mat.det.cofactor.col
GFDL-1.2
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September 11th, 2017
6 years ago
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# Determinant of a Matrix - A First Course in Linear Algebra

Before we define the determinant of a matrix, we take a slight detour to introduce elementary matrices. These will bring us back to the beginning of the course and our old friend, row operations.

• ##### A matrix and its transpose have the same determinant. math.la.t.mat.det.transpose
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September 11th, 2017
6 years ago
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# Properties of Determinants of Matrices - A First Course in Linear Algebra

We have seen how to compute the determinant of a matrix, and the incredible fact that we can perform expansion about any row or column to make this computation. In this largely theoretical section, we will state and prove several more intriguing properties about determinants. Our main goal will be the two results in Theorem SMZD and Theorem DRMM, but more specifically, we will see how the value of a determinant will allow us to gain insight into the various properties of a square matrix.

GFDL-1.2
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September 11th, 2017
6 years ago
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# Properties of Determinants of Matrices - A First Course in Linear Algebra

We have seen how to compute the determinant of a matrix, and the incredible fact that we can perform expansion about any row or column to make this computation. In this largely theoretical section, we will state and prove several more intriguing properties about determinants. Our main goal will be the two results in Theorem SMZD and Theorem DRMM, but more specifically, we will see how the value of a determinant will allow us to gain insight into the various properties of a square matrix.

• ##### math.la.t.mat.det.scalar
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
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English
Content Type
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# Properties of Determinants of Matrices - A First Course in Linear Algebra

We have seen how to compute the determinant of a matrix, and the incredible fact that we can perform expansion about any row or column to make this computation. In this largely theoretical section, we will state and prove several more intriguing properties about determinants. Our main goal will be the two results in Theorem SMZD and Theorem DRMM, but more specifically, we will see how the value of a determinant will allow us to gain insight into the various properties of a square matrix.

• ##### math.la.t.mat.row.z
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
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English
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# Properties of Determinants of Matrices - A First Course in Linear Algebra

We have seen how to compute the determinant of a matrix, and the incredible fact that we can perform expansion about any row or column to make this computation. In this largely theoretical section, we will state and prove several more intriguing properties about determinants. Our main goal will be the two results in Theorem SMZD and Theorem DRMM, but more specifically, we will see how the value of a determinant will allow us to gain insight into the various properties of a square matrix.

• ##### math.la.t.mat.det.switch
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
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# Properties of Determinants of Matrices - A First Course in Linear Algebra

We have seen how to compute the determinant of a matrix, and the incredible fact that we can perform expansion about any row or column to make this computation. In this largely theoretical section, we will state and prove several more intriguing properties about determinants. Our main goal will be the two results in Theorem SMZD and Theorem DRMM, but more specifically, we will see how the value of a determinant will allow us to gain insight into the various properties of a square matrix.

• ##### math.la.t.mat.row.equal
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
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# Properties of Determinants of Matrices - A First Course in Linear Algebra

We have seen how to compute the determinant of a matrix, and the incredible fact that we can perform expansion about any row or column to make this computation. In this largely theoretical section, we will state and prove several more intriguing properties about determinants. Our main goal will be the two results in Theorem SMZD and Theorem DRMM, but more specifically, we will see how the value of a determinant will allow us to gain insight into the various properties of a square matrix.

• ##### Theorem describing the determinants of elementary matrices. math.la.t.mat.elementary.det
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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# Determinants and row operations

The effect of row operations on the determinant of a matrix; computing determinants via row reduction; a square matrix is invertible if and only if its determinant is nonzero.

• ##### The determinant of a triangular matrix is the product of the entries on the diagonal. math.la.t.mat.det.trianglar
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August 22nd, 2017
6 years ago
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4
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# Determinants and their relation to column operations and products

Determinant of the transpose equals the determinant of the original matrix; rescaling a column rescales the determinant by the same factor; interchanging two columns changes the sign of the determinant; adding multiple of one column to another leaves determinant unchanged; determinant of the product of two matrices equals product of the two determinants

• ##### Theorem describing the effect of elementary row operations on the determinant of a matrix. math.la.t.mat.det.elementaryoperations
Created On
August 25th, 2017
6 years ago
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3
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Video
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English
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# Properties of Determinants of Matrices - A First Course in Linear Algebra

We have seen how to compute the determinant of a matrix, and the incredible fact that we can perform expansion about any row or column to make this computation. In this largely theoretical section, we will state and prove several more intriguing properties about determinants. Our main goal will be the two results in Theorem SMZD and Theorem DRMM, but more specifically, we will see how the value of a determinant will allow us to gain insight into the various properties of a square matrix.

• ##### Equivalence theorem: the determinant of A is nonzero. math.la.t.equiv.det
GFDL-1.2
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September 11th, 2017
6 years ago
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2
Type
Textbook
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# Properties of Determinants of Matrices - A First Course in Linear Algebra

We have seen how to compute the determinant of a matrix, and the incredible fact that we can perform expansion about any row or column to make this computation. In this largely theoretical section, we will state and prove several more intriguing properties about determinants. Our main goal will be the two results in Theorem SMZD and Theorem DRMM, but more specifically, we will see how the value of a determinant will allow us to gain insight into the various properties of a square matrix.

• ##### If A and B are n-by-n matrices, then det(AB)=det(A)det(B). math.la.t.mat.det.product
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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# Pre-class activity: Introduction to eigenvalues and eigenvectors

An introductory activity on eigenvalues and eigenvectors in which students do basic matrix-vector multiplication calculations to find whether given vectors are eigenvectors, to determine the eigenvalue corresponding to an eigenvector and to find an eigenvector corresponding to an eigenvalue. This activity is self-contained and does not require any previous experience with eigenvalues or eigenvectors.

• ##### Definition of eigenvalue(s) of a matrix math.la.d.mat.eig
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June 9th, 2017
6 years ago
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Introduction
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# Eigenvalues and Eigenvectors - A First Course in Linear Algebra

In this section, we will define the eigenvalues and eigenvectors of a matrix, and see how to compute them. More theoretical properties will be taken up in the next section.

• ##### Definition of eigenvalue(s) of a matrix math.la.d.mat.eig
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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# Geometry of eigenvectors

Definition of the eigenspace corresponding to an eigenvector $\lambda$ (and proof that this is a vector space); analysis of simple matrices in R^2 and R^3 to visualize the "geometry" of eigenspaces; proof that eigenvectors corresponding to distinct eigenvectors are linearly independent

• ##### Definition of eigenspace(s) of a matrix math.la.d.mat.eigsp
Created On
August 25th, 2017
6 years ago
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2
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Video
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English
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# Characteristic equation of a matrix

Theorem: \lambda is an eigenvalue of a matrix A if and only if \lambda satisfies the characteristic equation det(A-\lambda I) = 0; examples; eigenvalues of triangular matrices are the diagonal entries.

• ##### The eigenvalues of a triangular matrix are the entries on the main diagonal. math.la.t.mat.eig.triangular
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September 3rd, 2017
6 years ago
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2
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# Eigenvalues and Eigenvectors - A First Course in Linear Algebra

In this section, we will define the eigenvalues and eigenvectors of a matrix, and see how to compute them. More theoretical properties will be taken up in the next section.

• ##### math.la.t.mat.eigsp.subsp
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
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Textbook
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# Eigenvalues and Eigenvectors - A First Course in Linear Algebra

In this section, we will define the eigenvalues and eigenvectors of a matrix, and see how to compute them. More theoretical properties will be taken up in the next section.

• ##### Definition of eigenspace(s) of a matrix math.la.d.mat.eigsp
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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# Eigenvalues and Eigenvectors - A First Course in Linear Algebra

In this section, we will define the eigenvalues and eigenvectors of a matrix, and see how to compute them. More theoretical properties will be taken up in the next section.

• ##### Definition of characteristic polynomial of a matrix math.la.d.mat.charpoly
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September 11th, 2017
6 years ago
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2
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# Eigenvalues and Eigenvectors - A First Course in Linear Algebra

In this section, we will define the eigenvalues and eigenvectors of a matrix, and see how to compute them. More theoretical properties will be taken up in the next section.

• ##### math.la.t.mat.eigsp.nullspace
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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# Eigenvalues and Eigenvectors - A First Course in Linear Algebra

In this section, we will define the eigenvalues and eigenvectors of a matrix, and see how to compute them. More theoretical properties will be taken up in the next section.

• ##### The eigenvalues of a matrix are the roots/solutions of its characteristic polynomial/equation. math.la.t.mat.charpoly.eig
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September 11th, 2017
6 years ago
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3
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# Eigenvalues and Eigenvectors - A First Course in Linear Algebra

In this section, we will define the eigenvalues and eigenvectors of a matrix, and see how to compute them. More theoretical properties will be taken up in the next section.

• ##### math.la.d.mat.eig.multiplicity.geometric
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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# Eigenvalues and Eigenvectors - A First Course in Linear Algebra

In this section, we will define the eigenvalues and eigenvectors of a matrix, and see how to compute them. More theoretical properties will be taken up in the next section.

• ##### math.la.d.mat.eig.multiplicity.algebraic
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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# Geometry of eigenvectors

Definition of the eigenspace corresponding to an eigenvector $\lambda$ (and proof that this is a vector space); analysis of simple matrices in R^2 and R^3 to visualize the "geometry" of eigenspaces; proof that eigenvectors corresponding to distinct eigenvectors are linearly independent

• ##### Definition of eigenspace(s) of a matrix math.la.d.mat.eigsp
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August 25th, 2017
6 years ago
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2
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# Properties of Eigenvalues and Eigenvectors - A First Course in Linear Algebra

The previous section introduced eigenvalues and eigenvectors, and concentrated on their existence and determination. This section will be more about theorems, and the various properties eigenvalues and eigenvectors enjoy. Like a good 4×100 meter relay, we will lead-off with one of our better theorems and save the very best for the anchor leg.

• ##### Equivalence theorem: the matrix A does not have zero as an eigenvalue. math.la.t.equiv.eig
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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# Properties of Eigenvalues and Eigenvectors - A First Course in Linear Algebra

The previous section introduced eigenvalues and eigenvectors, and concentrated on their existence and determination. This section will be more about theorems, and the various properties eigenvalues and eigenvectors enjoy. Like a good 4×100 meter relay, we will lead-off with one of our better theorems and save the very best for the anchor leg.

• ##### math.la.t.mat.eig.inv
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
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Textbook
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English
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# Properties of Eigenvalues and Eigenvectors - A First Course in Linear Algebra

The previous section introduced eigenvalues and eigenvectors, and concentrated on their existence and determination. This section will be more about theorems, and the various properties eigenvalues and eigenvectors enjoy. Like a good 4×100 meter relay, we will lead-off with one of our better theorems and save the very best for the anchor leg.

• ##### math.la.t.mat.eig.scalar
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
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English
Content Type
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# Properties of Eigenvalues and Eigenvectors - A First Course in Linear Algebra

The previous section introduced eigenvalues and eigenvectors, and concentrated on their existence and determination. This section will be more about theorems, and the various properties eigenvalues and eigenvectors enjoy. Like a good 4×100 meter relay, we will lead-off with one of our better theorems and save the very best for the anchor leg.

• ##### math.la.t.mat.eig.polynomial
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
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Textbook
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English
Content Type
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# Properties of Eigenvalues and Eigenvectors - A First Course in Linear Algebra

The previous section introduced eigenvalues and eigenvectors, and concentrated on their existence and determination. This section will be more about theorems, and the various properties eigenvalues and eigenvectors enjoy. Like a good 4×100 meter relay, we will lead-off with one of our better theorems and save the very best for the anchor leg.

• ##### Eigenvectors with distinct eigenvalues are linearly independent. math.la.t.mat.eigvec.linindep
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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Textbook
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English
Content Type
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# Properties of Eigenvalues and Eigenvectors - A First Course in Linear Algebra

The previous section introduced eigenvalues and eigenvectors, and concentrated on their existence and determination. This section will be more about theorems, and the various properties eigenvalues and eigenvectors enjoy. Like a good 4×100 meter relay, we will lead-off with one of our better theorems and save the very best for the anchor leg.

• ##### A matrix with real entries has eigenvalues occurring in conjugate pairs. math.la.t.mat.real.eig.cn
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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Textbook
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# Properties of Eigenvalues and Eigenvectors - A First Course in Linear Algebra

The previous section introduced eigenvalues and eigenvectors, and concentrated on their existence and determination. This section will be more about theorems, and the various properties eigenvalues and eigenvectors enjoy. Like a good 4×100 meter relay, we will lead-off with one of our better theorems and save the very best for the anchor leg.

• ##### math.la.t.mat.eig.power
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
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Textbook
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English
Content Type
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# Properties of Eigenvalues and Eigenvectors - A First Course in Linear Algebra

The previous section introduced eigenvalues and eigenvectors, and concentrated on their existence and determination. This section will be more about theorems, and the various properties eigenvalues and eigenvectors enjoy. Like a good 4×100 meter relay, we will lead-off with one of our better theorems and save the very best for the anchor leg.

• ##### math.la.t.mat.eig.transpose
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
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English
Content Type
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# Properties of Eigenvalues and Eigenvectors - A First Course in Linear Algebra

The previous section introduced eigenvalues and eigenvectors, and concentrated on their existence and determination. This section will be more about theorems, and the various properties eigenvalues and eigenvectors enjoy. Like a good 4×100 meter relay, we will lead-off with one of our better theorems and save the very best for the anchor leg.

• ##### The dimension of a eigenspace is less than or equal to the multiplicity of the eigenvalue. math.la.t.mat.eig.multiplicity.eigenspace
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
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Textbook
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English
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# Properties of Eigenvalues and Eigenvectors - A First Course in Linear Algebra

The previous section introduced eigenvalues and eigenvectors, and concentrated on their existence and determination. This section will be more about theorems, and the various properties eigenvalues and eigenvectors enjoy. Like a good 4×100 meter relay, we will lead-off with one of our better theorems and save the very best for the anchor leg.

• ##### math.la.d.mat.eig.number
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
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# Properties of Eigenvalues and Eigenvectors - A First Course in Linear Algebra

The previous section introduced eigenvalues and eigenvectors, and concentrated on their existence and determination. This section will be more about theorems, and the various properties eigenvalues and eigenvectors enjoy. Like a good 4×100 meter relay, we will lead-off with one of our better theorems and save the very best for the anchor leg.

• ##### math.la.t.mat.hermitian.eigvec.orthogonal
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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2
Type
Textbook
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English
Content Type
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# Properties of Eigenvalues and Eigenvectors - A First Course in Linear Algebra

The previous section introduced eigenvalues and eigenvectors, and concentrated on their existence and determination. This section will be more about theorems, and the various properties eigenvalues and eigenvectors enjoy. Like a good 4×100 meter relay, we will lead-off with one of our better theorems and save the very best for the anchor leg.

• ##### math.la.t.mat.hermitian.eig.real
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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Textbook
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# Similarity of matrices

Definition of similarity for square matrices; similarity is an equivalence relation; similar matrices have the same characteristic polynomial and hence the same eigenvalues, with same multiplicities; definition of multiplicity.

• ##### Similar matrices have the same eigenvalues and the same characteristic polynomials. math.la.t.mat.similar.eig
Created On
September 3rd, 2017
6 years ago
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2
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Video
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English
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# Similarity and Diagonalization - A First Course in Linear Algebra

This section's topic will perhaps seem out of place at first, but we will make the connection soon with eigenvalues and eigenvectors. This is also our first look at one of the central ideas of Chapter R.

• ##### Definition of similar matrices math.la.d.mat.similar
GFDL-1.2
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September 11th, 2017
6 years ago
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# Similarity of matrices

Definition of similarity for square matrices; similarity is an equivalence relation; similar matrices have the same characteristic polynomial and hence the same eigenvalues, with same multiplicities; definition of multiplicity.

• ##### Similar matrices have the same eigenvalues and the same characteristic polynomials. math.la.t.mat.similar.eig
Created On
September 3rd, 2017
6 years ago
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2
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Video
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English
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# Similarity and Diagonalization - A First Course in Linear Algebra

This section's topic will perhaps seem out of place at first, but we will make the connection soon with eigenvalues and eigenvectors. This is also our first look at one of the central ideas of Chapter R.

• ##### math.la.t.mat.similar.equiv
GFDL-1.2
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September 11th, 2017
6 years ago
Views
2
Type
Textbook
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English
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# Similarity and Diagonalization - A First Course in Linear Algebra

This section's topic will perhaps seem out of place at first, but we will make the connection soon with eigenvalues and eigenvectors. This is also our first look at one of the central ideas of Chapter R.

• ##### Similar matrices have the same eigenvalues and the same characteristic polynomials. math.la.t.mat.similar.eig
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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Textbook
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# Notation for matrix entries, size of a matrix, etc

Notation for matrix entries, diagonal matrix, square matrix, identity matrix, and zero matrix.

• ##### Definition of zero matrix math.la.d.mat.zero
Created On
February 17th, 2017
6 years ago
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3
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Video
Timeframe
Pre-class
Perspective
Introduction
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English
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# Diagonalization of Real Symmetric Matrices

A real matrix $A$ is symmetric if and only if it is orthogonally diagonalizable (i.e. $A = PDP^{-1}$ for an orthogonal matrix $P$.) Proof and examples.

• ##### The spectral theorem for symmetric matrices math.la.t.mat.symmetric.spectral
Created On
August 21st, 2017
6 years ago
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4
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Video
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English
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# Diagonalization of a matrix

Diagonalization theorem: a nxn matrix A is diagonalizable if and only if it has n linearly independent eigenvectors. If so, the matrix factors as A = PDP^{-1}, where D is diagonal and P is invertible (and its columns are the n linearly independent eigenvectors). Algorithm to diagonalize a matrix.

• ##### An n-by-n matrix with n distinct eigenvalues is diagonalizable. math.la.t.mat.diagonalizable.distinct
Created On
August 25th, 2017
6 years ago
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3
Type
Video
Language
English
Content Type
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# Similarity and Diagonalization - A First Course in Linear Algebra

This section's topic will perhaps seem out of place at first, but we will make the connection soon with eigenvalues and eigenvectors. This is also our first look at one of the central ideas of Chapter R.

• ##### An n-by-n matrix is diagonalizable if and only if it has n linearly independent eigenvectors. math.la.t.mat.diagonalizable
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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2
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Textbook
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English
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# Similarity and Diagonalization - A First Course in Linear Algebra

This section's topic will perhaps seem out of place at first, but we will make the connection soon with eigenvalues and eigenvectors. This is also our first look at one of the central ideas of Chapter R.

• ##### An n-by-n matrix with n distinct eigenvalues is diagonalizable. math.la.t.mat.diagonalizable.distinct
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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2
Type
Textbook
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English
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# Similarity and Diagonalization - A First Course in Linear Algebra

This section's topic will perhaps seem out of place at first, but we will make the connection soon with eigenvalues and eigenvectors. This is also our first look at one of the central ideas of Chapter R.

• ##### An n-by-n matrix is diagonalizable if and only if the characteristic polynomial factors completely, and the dimension of each eigenspace equals the multiplicity of the eigenvalue. math.la.t.mat.diagonalizable.charpoly
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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Textbook
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English
Content Type
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# Similarity and Diagonalization - A First Course in Linear Algebra

This section's topic will perhaps seem out of place at first, but we will make the connection soon with eigenvalues and eigenvectors. This is also our first look at one of the central ideas of Chapter R.

• ##### Definition of diagonal matrix math.la.d.mat.diagonal
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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Textbook
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English
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# Similarity and Diagonalization - A First Course in Linear Algebra

This section's topic will perhaps seem out of place at first, but we will make the connection soon with eigenvalues and eigenvectors. This is also our first look at one of the central ideas of Chapter R.

• ##### Definition of matrix diagonalization math.la.d.mat.diagonalization
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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# Why an abstract linear transformation maps 0 to 0

Two proofs, with discussion, of the fact that an abstract linear transformation maps 0 to 0.

• ##### Definition of linear transformation, arbitrary vector space math.la.d.lintrans.arb
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February 15th, 2017
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# Linear transformations and matrices | Essence of linear algebra, chapter 3

Matrices can be thought of as transforming space, and understanding how this work is crucial for understanding many other ideas that follow in linear algebra...

• ##### Definition of linear transformation, arbitrary vector space math.la.d.lintrans.arb
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# Lagrange Interpolation - Ximera

This is a guided discovery of the formula for Lagrange Interpolation, which lets you find the formula for a polynomial which passes through a given set of points.

• ##### Definition of linear transformation, arbitrary vector space math.la.d.lintrans.arb
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June 8th, 2017
6 years ago
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# Column space of a matrix

Definition of the column space of a matrix; column space is a subspace; comparison to the null space; definition of a linear transformation between vector spaces; definition of kernel and range of a linear transformation

• ##### Definition of range of linear transformation, arbitrary vector space math.la.d.lintrans.range.arb
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# Linear Transformations - A First Course in Linear Algebra

Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

• ##### A linear transformation maps zero to zero. math.la.t.lintrans.z
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6 years ago
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# Linear Transformations - A First Course in Linear Algebra

Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

• ##### Definition of linear transformation, arbitrary vector space math.la.d.lintrans.arb
GFDL-1.2
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# Pre-class quiz on Linear Transformations

After watching a video defining linear transformations and giving examples of 2-D transformations, students should be able to answer the questions in this quiz.

• ##### A linear transformation is given by a matrix whose columns are the images of the standard basis vectors, coordinate setting. math.la.t.lintrans.mat.basis.standard.coord
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June 8th, 2017
6 years ago
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# Linear transformations

Use matrix transformations to motivate the concept of linear transformation; examples of matrix transformations

• ##### Example of a linear transformation on R^2: projection math.la.e.lintrans.projection.r2
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# Linear Transformations - A First Course in Linear Algebra

Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

• ##### A linear transformation is given by a matrix whose columns are the images of the standard basis vectors, coordinate setting. math.la.t.lintrans.mat.basis.standard.coord
GFDL-1.2
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September 11th, 2017
6 years ago
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# Linear Transformations - A First Course in Linear Algebra

Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

• ##### Matrices act as a transformations by multiplying vectors math.la.c.transformation.matrix
GFDL-1.2
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September 11th, 2017
6 years ago
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# Linear Transformations - A First Course in Linear Algebra

Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

• ##### A linear transformation of a linear combination is the linear combination of the linear transformation math.la.t.lintrans.lincomb
GFDL-1.2
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6 years ago
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# Linear Transformations - A First Course in Linear Algebra

Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

• ##### math.la.t.lintrans.basis
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
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Textbook
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# Linear Transformations - A First Course in Linear Algebra

Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

• ##### Definition of pre-image of linear transformation, arbitrary vector space math.la.d.lintrans.preimage.arb
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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Textbook
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# Linear Transformations - A First Course in Linear Algebra

Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

• ##### math.la.d.lintrans.scalar.arb
GFDL-1.2
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September 11th, 2017
6 years ago
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2
Type
Textbook
Language
English
Content Type
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# Linear Transformations - A First Course in Linear Algebra

Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

• ##### math.la.t.lintrans.sum.arb
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
Type
Textbook
Language
English
Content Type
text/html

# Linear Transformations - A First Course in Linear Algebra

Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

• ##### math.la.t.lintrans.scalar.arb
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

# Linear Transformations - A First Course in Linear Algebra

Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

• ##### math.la.d.lintrans.sum.arb
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

# Linear Transformations - A First Course in Linear Algebra

Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

• ##### math.la.t.lintrans.composition.arb
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

# Linear Transformations - A First Course in Linear Algebra

Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

• ##### math.la.d.lintrans.composition.arb
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

# Linear Transformations - A First Course in Linear Algebra

Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

• ##### math.la.t.lintrans.vsp
GFDL-1.2
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September 11th, 2017
6 years ago
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3
Type
Textbook
Language
English
Content Type
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# Injective Linear Transformations - A First Course in Linear Algebra

Some linear transformations possess one, or both, of two key properties, which go by the names injective and surjective. We will see that they are closely related to ideas like linear independence and spanning, and subspaces like the null space and the column space. In this section we will define an injective linear transformation and analyze the resulting consequences. The next section will do the same for the surjective property. In the final section of this chapter we will see what happens when we have the two properties simultaneously.

• ##### Definition of one-to-one/injective linear transformation, arbitrary vector space math.la.d.lintrans.injective.arb
GFDL-1.2
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6 years ago
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# Column space of a matrix

Definition of the column space of a matrix; column space is a subspace; comparison to the null space; definition of a linear transformation between vector spaces; definition of kernel and range of a linear transformation

• ##### Definition of range of linear transformation, arbitrary vector space math.la.d.lintrans.range.arb
Created On
August 25th, 2017
6 years ago
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3
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Video
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# Injective Linear Transformations - A First Course in Linear Algebra

Some linear transformations possess one, or both, of two key properties, which go by the names injective and surjective. We will see that they are closely related to ideas like linear independence and spanning, and subspaces like the null space and the column space. In this section we will define an injective linear transformation and analyze the resulting consequences. The next section will do the same for the surjective property. In the final section of this chapter we will see what happens when we have the two properties simultaneously.

• ##### Definition of kernel of linear transformation, arbitrary vector space math.la.d.lintrans.kernel.arb
GFDL-1.2
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6 years ago
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3
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# Injective Linear Transformations - A First Course in Linear Algebra

Some linear transformations possess one, or both, of two key properties, which go by the names injective and surjective. We will see that they are closely related to ideas like linear independence and spanning, and subspaces like the null space and the column space. In this section we will define an injective linear transformation and analyze the resulting consequences. The next section will do the same for the surjective property. In the final section of this chapter we will see what happens when we have the two properties simultaneously.

• ##### math.la.t.lintrans.equiv.kernel
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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Textbook
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# Injective Linear Transformations - A First Course in Linear Algebra

Some linear transformations possess one, or both, of two key properties, which go by the names injective and surjective. We will see that they are closely related to ideas like linear independence and spanning, and subspaces like the null space and the column space. In this section we will define an injective linear transformation and analyze the resulting consequences. The next section will do the same for the surjective property. In the final section of this chapter we will see what happens when we have the two properties simultaneously.

• ##### math.la.t.lintrans.preimage.translation.arb
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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2
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Textbook
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English
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# Injective Linear Transformations - A First Course in Linear Algebra

Some linear transformations possess one, or both, of two key properties, which go by the names injective and surjective. We will see that they are closely related to ideas like linear independence and spanning, and subspaces like the null space and the column space. In this section we will define an injective linear transformation and analyze the resulting consequences. The next section will do the same for the surjective property. In the final section of this chapter we will see what happens when we have the two properties simultaneously.

• ##### math.la.t.lintrans.kernel.arb
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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2
Type
Textbook
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Content Type
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# Injective Linear Transformations - A First Course in Linear Algebra

Some linear transformations possess one, or both, of two key properties, which go by the names injective and surjective. We will see that they are closely related to ideas like linear independence and spanning, and subspaces like the null space and the column space. In this section we will define an injective linear transformation and analyze the resulting consequences. The next section will do the same for the surjective property. In the final section of this chapter we will see what happens when we have the two properties simultaneously.

• ##### math.la.t.lintrans.equiv.basis
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
Type
Textbook
Language
English
Content Type
text/html

# Injective Linear Transformations - A First Course in Linear Algebra

Some linear transformations possess one, or both, of two key properties, which go by the names injective and surjective. We will see that they are closely related to ideas like linear independence and spanning, and subspaces like the null space and the column space. In this section we will define an injective linear transformation and analyze the resulting consequences. The next section will do the same for the surjective property. In the final section of this chapter we will see what happens when we have the two properties simultaneously.

• ##### A linear transformation is one-to-one/injective if and only if the columns of its matrix are linearly independent. math.la.t.lintrans.injective.linindep
GFDL-1.2
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6 years ago
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3
Type
Textbook
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# Injective Linear Transformations - A First Course in Linear Algebra

Some linear transformations possess one, or both, of two key properties, which go by the names injective and surjective. We will see that they are closely related to ideas like linear independence and spanning, and subspaces like the null space and the column space. In this section we will define an injective linear transformation and analyze the resulting consequences. The next section will do the same for the surjective property. In the final section of this chapter we will see what happens when we have the two properties simultaneously.

• ##### math.la.t.lintrans.injective.dim
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

# Injective Linear Transformations - A First Course in Linear Algebra

Some linear transformations possess one, or both, of two key properties, which go by the names injective and surjective. We will see that they are closely related to ideas like linear independence and spanning, and subspaces like the null space and the column space. In this section we will define an injective linear transformation and analyze the resulting consequences. The next section will do the same for the surjective property. In the final section of this chapter we will see what happens when we have the two properties simultaneously.

• ##### math.la.t.lintrans.composition.injective.arb
GFDL-1.2
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September 11th, 2017
6 years ago
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2
Type
Textbook
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English
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# Surjective Linear Transformations - A First Course in Linear Algebra

The companion to an injection is a surjection. Surjective linear transformations are closely related to spanning sets and ranges. So as you read this section reflect back on Section ILT and note the parallels and the contrasts. In the next section, Section IVLT, we will combine the two properties.

• ##### Definition of onto/surjective linear transformation, arbitrary vector space math.la.d.lintrans.surjective.arb
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6 years ago
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# Column space of a matrix

Definition of the column space of a matrix; column space is a subspace; comparison to the null space; definition of a linear transformation between vector spaces; definition of kernel and range of a linear transformation

• ##### Definition of range of linear transformation, arbitrary vector space math.la.d.lintrans.range.arb
Created On
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6 years ago
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3
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Video
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English
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# Surjective Linear Transformations - A First Course in Linear Algebra

The companion to an injection is a surjection. Surjective linear transformations are closely related to spanning sets and ranges. So as you read this section reflect back on Section ILT and note the parallels and the contrasts. In the next section, Section IVLT, we will combine the two properties.

• ##### Definition of range of linear transformation, arbitrary vector space math.la.d.lintrans.range.arb
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
Type
Textbook
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English
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# Surjective Linear Transformations - A First Course in Linear Algebra

The companion to an injection is a surjection. Surjective linear transformations are closely related to spanning sets and ranges. So as you read this section reflect back on Section ILT and note the parallels and the contrasts. In the next section, Section IVLT, we will combine the two properties.

• ##### math.la.t.lintrans.range.subsp.arb
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
Type
Textbook
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English
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# Surjective Linear Transformations - A First Course in Linear Algebra

The companion to an injection is a surjection. Surjective linear transformations are closely related to spanning sets and ranges. So as you read this section reflect back on Section ILT and note the parallels and the contrasts. In the next section, Section IVLT, we will combine the two properties.

• ##### math.la.t.lintrans.basis.span.surjective.arb
GFDL-1.2
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September 11th, 2017
6 years ago
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2
Type
Textbook
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# Surjective Linear Transformations - A First Course in Linear Algebra

The companion to an injection is a surjection. Surjective linear transformations are closely related to spanning sets and ranges. So as you read this section reflect back on Section ILT and note the parallels and the contrasts. In the next section, Section IVLT, we will combine the two properties.

• ##### math.la.t.lintrans.range.span.arb
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

# Surjective Linear Transformations - A First Course in Linear Algebra

The companion to an injection is a surjection. Surjective linear transformations are closely related to spanning sets and ranges. So as you read this section reflect back on Section ILT and note the parallels and the contrasts. In the next section, Section IVLT, we will combine the two properties.

• ##### math.la.t.lintrans.surjective.dim
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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2
Type
Textbook
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English
Content Type
text/html

# Surjective Linear Transformations - A First Course in Linear Algebra

The companion to an injection is a surjection. Surjective linear transformations are closely related to spanning sets and ranges. So as you read this section reflect back on Section ILT and note the parallels and the contrasts. In the next section, Section IVLT, we will combine the two properties.

• ##### math.la.t.lintrans.composition.surjective.arb
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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Textbook
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# Invertible Linear Transformations - A First Course in Linear Algebra

In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

• ##### math.la.t.lintrans.inv.involution.arb
GFDL-1.2
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6 years ago
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Textbook
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# Invertible Linear Transformations - A First Course in Linear Algebra

In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

• ##### math.la.d.lintrans.inv.arb
GFDL-1.2
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September 11th, 2017
6 years ago
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# Invertible Linear Transformations - A First Course in Linear Algebra

In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

• ##### math.la.d.lintrans.identity
GFDL-1.2
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September 11th, 2017
6 years ago
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# Invertible Linear Transformations - A First Course in Linear Algebra

In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

• ##### math.la.t.lintrans.inv.arb
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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Textbook
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English
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# Invertible Linear Transformations - A First Course in Linear Algebra

In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

• ##### math.la.t.lintrans.composition.invertible.arb
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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# Invertible Linear Transformations - A First Course in Linear Algebra

In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

• ##### math.la.t.lintrans.inv.shoesandsocks
GFDL-1.2
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September 11th, 2017
6 years ago
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# Invertible Linear Transformations - A First Course in Linear Algebra

In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

• ##### math.la.t.lintrans.invertible.arb
GFDL-1.2
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6 years ago
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3
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# Coordinate systems and isomorphic vector spaces

Given a basis for a n-dimensional vector space V, the coordinate map is a linear bijection between V and R^n; definition isomorphisms between vector spaces and isomorphic vector spaces.

• ##### The coordinate vector/mapping relative to a given basis is a bijective linear mapping to R^n (or C^n). math.la.t.vsp.basis.coord.vector.arb
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6 years ago
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# Invertible Linear Transformations - A First Course in Linear Algebra

In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

• ##### Definition of isomorphism between vector spaces math.la.d.vsp.isomorphism
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# Vector Representations - A First Course in Linear Algebra

You may have noticed that many questions about elements of abstract vector spaces eventually become questions about column vectors or systems of equations. Example SM32 would be an example of this. We will make this vague idea more precise in this section.

• ##### math.la.t.vsp.dim.isomorphic
GFDL-1.2
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6 years ago
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# Invertible Linear Transformations - A First Course in Linear Algebra

In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

• ##### math.la.t.vsp.isomorphic.dim
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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Textbook
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# Invertible Linear Transformations - A First Course in Linear Algebra

In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

• ##### math.la.t.lintrans.equiv.nullity
GFDL-1.2
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September 11th, 2017
6 years ago
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2
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# Invertible Linear Transformations - A First Course in Linear Algebra

In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

• ##### math.la.t.lintrans.ranknullity
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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Textbook
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English
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# Invertible Linear Transformations - A First Course in Linear Algebra

In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

• ##### math.la.t.lintrans.surjective.rank
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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Textbook
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English
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# Invertible Linear Transformations - A First Course in Linear Algebra

In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

• ##### Definition of rank of a linear transformation math.la.d.lintrans.rank
GFDL-1.2
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September 11th, 2017
6 years ago
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Textbook
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# Invertible Linear Transformations - A First Course in Linear Algebra

In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

• ##### Definition of nullity of a linear transformation math.la.d.lintrans.nullity
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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Textbook
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# Vector Representations - A First Course in Linear Algebra

You may have noticed that many questions about elements of abstract vector spaces eventually become questions about column vectors or systems of equations. Example SM32 would be an example of this. We will make this vague idea more precise in this section.

• ##### Definition of coordinate vector/mapping relative to a given basis, arbitrary vector space math.la.d.vsp.basis.coord.vector.arb
GFDL-1.2
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September 11th, 2017
6 years ago
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2
Type
Textbook
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English
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# Vector Representations - A First Course in Linear Algebra

You may have noticed that many questions about elements of abstract vector spaces eventually become questions about column vectors or systems of equations. Example SM32 would be an example of this. We will make this vague idea more precise in this section.

• ##### math.la.t.vsp.basis.coord.lin.arb
GFDL-1.2
Submitted At
September 11th, 2017
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# Vector Representations - A First Course in Linear Algebra

You may have noticed that many questions about elements of abstract vector spaces eventually become questions about column vectors or systems of equations. Example SM32 would be an example of this. We will make this vague idea more precise in this section.

• ##### math.la.t.vsp.basis.coord.surjective.arb
GFDL-1.2
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# Vector Representations - A First Course in Linear Algebra

You may have noticed that many questions about elements of abstract vector spaces eventually become questions about column vectors or systems of equations. Example SM32 would be an example of this. We will make this vague idea more precise in this section.

• ##### math.la.t.vsp.basis.coord.injective.arb
GFDL-1.2
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September 11th, 2017
6 years ago
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# Vector Representations - A First Course in Linear Algebra

You may have noticed that many questions about elements of abstract vector spaces eventually become questions about column vectors or systems of equations. Example SM32 would be an example of this. We will make this vague idea more precise in this section.

• ##### math.la.t.vsp.dim.isomorphic
GFDL-1.2
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6 years ago
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# Invertible Linear Transformations - A First Course in Linear Algebra

In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

• ##### math.la.t.vsp.isomorphic.dim
GFDL-1.2
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# Vector Representations - A First Course in Linear Algebra

You may have noticed that many questions about elements of abstract vector spaces eventually become questions about column vectors or systems of equations. Example SM32 would be an example of this. We will make this vague idea more precise in this section.

• ##### math.la.t.vsp.isomorphic.rncn
GFDL-1.2
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September 11th, 2017
6 years ago
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# Vector Representations - A First Course in Linear Algebra

You may have noticed that many questions about elements of abstract vector spaces eventually become questions about column vectors or systems of equations. Example SM32 would be an example of this. We will make this vague idea more precise in this section.

• ##### math.la.t.vsp.linindep.coord
GFDL-1.2
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6 years ago
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# Matrix Representations - A First Course in Linear Algebra

We have seen that linear transformations whose domain and codomain are vector spaces of columns vectors have a close relationship with matrices (Theorem MBLT, Theorem MLTCV). In this section, we will extend the relationship between matrices and linear transformations to the setting of linear transformations between abstract vector spaces.

• ##### math.la.t.lintrans.mat.repn.arb
GFDL-1.2
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September 11th, 2017
6 years ago
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# Matrix Representations - A First Course in Linear Algebra

We have seen that linear transformations whose domain and codomain are vector spaces of columns vectors have a close relationship with matrices (Theorem MBLT, Theorem MLTCV). In this section, we will extend the relationship between matrices and linear transformations to the setting of linear transformations between abstract vector spaces.

• ##### Definition of matrix representation of a linear transformation with respect to bases of the spaces, arbitrary vector space math.la.d.lintrans.mat.repn.arb
GFDL-1.2
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6 years ago
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# Matrix Representations - A First Course in Linear Algebra

We have seen that linear transformations whose domain and codomain are vector spaces of columns vectors have a close relationship with matrices (Theorem MBLT, Theorem MLTCV). In this section, we will extend the relationship between matrices and linear transformations to the setting of linear transformations between abstract vector spaces.

• ##### math.la.t.lintrans.mat_repn.composition
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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2
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# Matrix Representations - A First Course in Linear Algebra

We have seen that linear transformations whose domain and codomain are vector spaces of columns vectors have a close relationship with matrices (Theorem MBLT, Theorem MLTCV). In this section, we will extend the relationship between matrices and linear transformations to the setting of linear transformations between abstract vector spaces.

• ##### math.la.t.lintrans.mat_repn.sum
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
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# Matrix Representations - A First Course in Linear Algebra

We have seen that linear transformations whose domain and codomain are vector spaces of columns vectors have a close relationship with matrices (Theorem MBLT, Theorem MLTCV). In this section, we will extend the relationship between matrices and linear transformations to the setting of linear transformations between abstract vector spaces.

• ##### math.la.t.lintrans.mat_repn.scalar
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
Type
Textbook
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Content Type
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# Matrix Representations - A First Course in Linear Algebra

We have seen that linear transformations whose domain and codomain are vector spaces of columns vectors have a close relationship with matrices (Theorem MBLT, Theorem MLTCV). In this section, we will extend the relationship between matrices and linear transformations to the setting of linear transformations between abstract vector spaces.

• ##### Equivalence theorem: the linear transformation given by T(x)=Ax has an inverse. math.la.t.equiv.lintrans.inv
GFDL-1.2
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6 years ago
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# Matrix Representations - A First Course in Linear Algebra

We have seen that linear transformations whose domain and codomain are vector spaces of columns vectors have a close relationship with matrices (Theorem MBLT, Theorem MLTCV). In this section, we will extend the relationship between matrices and linear transformations to the setting of linear transformations between abstract vector spaces.

• ##### math.la.t.lintrans.mat_repn.inv
GFDL-1.2
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6 years ago
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# Change of Basis - A First Course in Linear Algebra

We have seen in Section MR that a linear transformation can be represented by a matrix, once we pick bases for the domain and codomain. How does the matrix representation change if we choose different bases? Which bases lead to especially nice representations? From the infinite possibilities, what is the best possible representation? This section will begin to answer these questions. But first we need to define eigenvalues for linear transformations and the change-of-basis matrix.

• ##### math.la.d.lintrans.eigvec
GFDL-1.2
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# Lagrange Interpolation - Ximera

This is a guided discovery of the formula for Lagrange Interpolation, which lets you find the formula for a polynomial which passes through a given set of points.

• ##### Definition of linear transformation, arbitrary vector space math.la.d.lintrans.arb
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# Change of Basis - A First Course in Linear Algebra

We have seen in Section MR that a linear transformation can be represented by a matrix, once we pick bases for the domain and codomain. How does the matrix representation change if we choose different bases? Which bases lead to especially nice representations? From the infinite possibilities, what is the best possible representation? This section will begin to answer these questions. But first we need to define eigenvalues for linear transformations and the change-of-basis matrix.

• ##### math.la.t.vsp.change_of_basis
GFDL-1.2
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6 years ago
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2
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# Change of Basis - A First Course in Linear Algebra

We have seen in Section MR that a linear transformation can be represented by a matrix, once we pick bases for the domain and codomain. How does the matrix representation change if we choose different bases? Which bases lead to especially nice representations? From the infinite possibilities, what is the best possible representation? This section will begin to answer these questions. But first we need to define eigenvalues for linear transformations and the change-of-basis matrix.

• ##### Definition of change of corrdinates matrix between two bases, arbitrary vector space math.la.d.vsp.change_of_basis.arb
GFDL-1.2
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# Change of Basis - A First Course in Linear Algebra

We have seen in Section MR that a linear transformation can be represented by a matrix, once we pick bases for the domain and codomain. How does the matrix representation change if we choose different bases? Which bases lead to especially nice representations? From the infinite possibilities, what is the best possible representation? This section will begin to answer these questions. But first we need to define eigenvalues for linear transformations and the change-of-basis matrix.

• ##### math.la.t.vsp.change_of_basis.inv
GFDL-1.2
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September 11th, 2017
6 years ago
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# Change of Basis - A First Course in Linear Algebra

We have seen in Section MR that a linear transformation can be represented by a matrix, once we pick bases for the domain and codomain. How does the matrix representation change if we choose different bases? Which bases lead to especially nice representations? From the infinite possibilities, what is the best possible representation? This section will begin to answer these questions. But first we need to define eigenvalues for linear transformations and the change-of-basis matrix.

• ##### math.la.t.lintrans.mat_repn.eig
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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2
Type
Textbook
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Content Type
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# Change of Basis - A First Course in Linear Algebra

We have seen in Section MR that a linear transformation can be represented by a matrix, once we pick bases for the domain and codomain. How does the matrix representation change if we choose different bases? Which bases lead to especially nice representations? From the infinite possibilities, what is the best possible representation? This section will begin to answer these questions. But first we need to define eigenvalues for linear transformations and the change-of-basis matrix.

• ##### math.la.t.vsp.change_of_basis.conjugate
GFDL-1.2
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September 11th, 2017
6 years ago
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# Orthonormal Diagonalization - A First Course in Linear Algebra

We have seen in Section SD that under the right conditions a square matrix is similar to a diagonal matrix. We recognize now, via Theorem SCB, that a similarity transformation is a change of basis on a matrix representation. So we can now discuss the choice of a basis used to build a matrix representation, and decide if some bases are better than others for this purpose. This will be the tone of this section. We will also see that every matrix has a reasonably useful matrix representation, and we will discover a new class of diagonalizable linear transformations. First we need some basic facts about triangular matrices.

• ##### math.la.t.mat.triangular.prod
GFDL-1.2
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6 years ago
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# Orthonormal Diagonalization - A First Course in Linear Algebra

We have seen in Section SD that under the right conditions a square matrix is similar to a diagonal matrix. We recognize now, via Theorem SCB, that a similarity transformation is a change of basis on a matrix representation. So we can now discuss the choice of a basis used to build a matrix representation, and decide if some bases are better than others for this purpose. This will be the tone of this section. We will also see that every matrix has a reasonably useful matrix representation, and we will discover a new class of diagonalizable linear transformations. First we need some basic facts about triangular matrices.

• ##### math.la.t.mat.triangular.inv
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
Type
Textbook
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English
Content Type
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# Orthonormal Diagonalization - A First Course in Linear Algebra

We have seen in Section SD that under the right conditions a square matrix is similar to a diagonal matrix. We recognize now, via Theorem SCB, that a similarity transformation is a change of basis on a matrix representation. So we can now discuss the choice of a basis used to build a matrix representation, and decide if some bases are better than others for this purpose. This will be the tone of this section. We will also see that every matrix has a reasonably useful matrix representation, and we will discover a new class of diagonalizable linear transformations. First we need some basic facts about triangular matrices.

• ##### Definition of an upper triangular matrix math.la.d.mat.triangular.upper
GFDL-1.2
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September 11th, 2017
6 years ago
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2
Type
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# Orthonormal Diagonalization - A First Course in Linear Algebra

We have seen in Section SD that under the right conditions a square matrix is similar to a diagonal matrix. We recognize now, via Theorem SCB, that a similarity transformation is a change of basis on a matrix representation. So we can now discuss the choice of a basis used to build a matrix representation, and decide if some bases are better than others for this purpose. This will be the tone of this section. We will also see that every matrix has a reasonably useful matrix representation, and we will discover a new class of diagonalizable linear transformations. First we need some basic facts about triangular matrices.

• ##### Definition of an upper triangular matrix math.la.d.mat.triangular.upper
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
Type
Textbook
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English
Content Type
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# Orthonormal Diagonalization - A First Course in Linear Algebra

We have seen in Section SD that under the right conditions a square matrix is similar to a diagonal matrix. We recognize now, via Theorem SCB, that a similarity transformation is a change of basis on a matrix representation. So we can now discuss the choice of a basis used to build a matrix representation, and decide if some bases are better than others for this purpose. This will be the tone of this section. We will also see that every matrix has a reasonably useful matrix representation, and we will discover a new class of diagonalizable linear transformations. First we need some basic facts about triangular matrices.

• ##### math.la.t.lintrans.mat_repn.triangular
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

# Orthonormal Diagonalization - A First Course in Linear Algebra

We have seen in Section SD that under the right conditions a square matrix is similar to a diagonal matrix. We recognize now, via Theorem SCB, that a similarity transformation is a change of basis on a matrix representation. So we can now discuss the choice of a basis used to build a matrix representation, and decide if some bases are better than others for this purpose. This will be the tone of this section. We will also see that every matrix has a reasonably useful matrix representation, and we will discover a new class of diagonalizable linear transformations. First we need some basic facts about triangular matrices.

• ##### math.la.t.mat.triangular.unitary
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
Type
Textbook
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English
Content Type
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# Orthonormal Diagonalization - A First Course in Linear Algebra

We have seen in Section SD that under the right conditions a square matrix is similar to a diagonal matrix. We recognize now, via Theorem SCB, that a similarity transformation is a change of basis on a matrix representation. So we can now discuss the choice of a basis used to build a matrix representation, and decide if some bases are better than others for this purpose. This will be the tone of this section. We will also see that every matrix has a reasonably useful matrix representation, and we will discover a new class of diagonalizable linear transformations. First we need some basic facts about triangular matrices.

• ##### math.la.d.mat.normal
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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2
Type
Textbook
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English
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# Orthonormal Diagonalization - A First Course in Linear Algebra

We have seen in Section SD that under the right conditions a square matrix is similar to a diagonal matrix. We recognize now, via Theorem SCB, that a similarity transformation is a change of basis on a matrix representation. So we can now discuss the choice of a basis used to build a matrix representation, and decide if some bases are better than others for this purpose. This will be the tone of this section. We will also see that every matrix has a reasonably useful matrix representation, and we will discover a new class of diagonalizable linear transformations. First we need some basic facts about triangular matrices.

• ##### math.la.t.mat.normal.diagonalize
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

# Orthonormal Diagonalization - A First Course in Linear Algebra

We have seen in Section SD that under the right conditions a square matrix is similar to a diagonal matrix. We recognize now, via Theorem SCB, that a similarity transformation is a change of basis on a matrix representation. So we can now discuss the choice of a basis used to build a matrix representation, and decide if some bases are better than others for this purpose. This will be the tone of this section. We will also see that every matrix has a reasonably useful matrix representation, and we will discover a new class of diagonalizable linear transformations. First we need some basic facts about triangular matrices.