# 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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# 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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# Inconsistent linear systems

The reduced row echelon form is used to determine when a 3x3 system is inconsistent. A picture of planes in 3-dimensional space is used to provide geometric intuition.

• ##### Example of solving a 3-by-3 system of linear equations by row-reducing the augmented matrix, in the case of no solutions math.la.e.linsys.3x3.soln.row_reduce.z
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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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# 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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# 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
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August 21st, 2017
5 years ago
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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
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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
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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
GFDL-1.2
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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
GFDL-1.2
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# 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
GFDL-1.2
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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
GFDL-1.2
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5 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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5 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
GFDL-1.2
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September 11th, 2017
5 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 row operations on a matrix math.la.d.mat.row_op
GFDL-1.2
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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 coefficient matrix of a linear system math.la.d.mat.coeff
GFDL-1.2
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5 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
GFDL-1.2
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5 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 equivalent matrices math.la.d.mat.row_equiv
GFDL-1.2
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5 years ago
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2
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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
5 years ago
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3
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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
Submitted At
September 11th, 2017
5 years ago
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3
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Textbook
Language
English
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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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September 11th, 2017
5 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
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3
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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
5 years ago
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2
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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
5 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
5 years ago
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3
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Textbook
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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
Submitted At
September 11th, 2017
5 years ago
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2
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Textbook
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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
5 years ago
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3
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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
Created On
February 15th, 2017
6 years ago
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3
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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
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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
Created On
February 20th, 2017
6 years ago
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# 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
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February 20th, 2017
6 years ago
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# 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
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February 20th, 2017
6 years ago
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# Linear Combinations and Span

In-class activity for linear combinations and span.

GFDL-1.3
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June 8th, 2017
5 years ago
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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
Created On
October 23rd, 2013
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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October 23rd, 2013
9 years ago
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# 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
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October 23rd, 2013
9 years ago
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# 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
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3
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# 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
9 years ago
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4
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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
Created On
September 3rd, 2017
5 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 entry/component of a vector, coordinate vector space math.la.d.vec.component.coord
GFDL-1.2
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September 11th, 2017
5 years ago
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2
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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.

• ##### Definition of vector sum/addition, coordinate vector space math.la.d.vec.sum.coord
GFDL-1.2
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September 11th, 2017
5 years ago
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3
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# 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
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September 11th, 2017
5 years ago
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3
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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.

• ##### Definition of vector-scalar multiplication, coordinate vector space math.la.d.vec.scalar.mult.coord
GFDL-1.2
Submitted At
September 11th, 2017
5 years ago
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3
Type
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English
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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.

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

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

# 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
Created On
February 15th, 2017
6 years ago
Views
3
Type
Video
Timeframe
Pre-class
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Introduction
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English
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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
Created On
February 15th, 2017
6 years ago
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3
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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
Created On
February 15th, 2017
6 years ago
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# Solving Ax=0

For a specific 3x3 matrix, solve Ax=0 by row reducing an augmented matrix.

• ##### Example of solving a 3-by-3 homogeneous matrix equation math.la.e.mat.eqn.3x3.homog.solve
Created On
February 15th, 2017
6 years ago
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3
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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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Video
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Review
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English
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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
Created On
February 20th, 2017
6 years ago
Views
2
Type
Video
Timeframe
Pre-class
Perspective
Introduction
Language
English
Content Type
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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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# 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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# 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
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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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# 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
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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.

• ##### Matrix-scalar product is commutative math.la.t.mat.scalar.prod.commut
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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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# 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
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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
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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
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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
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# 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
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# Linear Independence in-class activity

Linear independence in-class activity

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# 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
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# 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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# 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
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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
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# 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
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# 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
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# 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
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# Definition of Linear Transformation

Linear transformations are defined, and some small examples (and non examples) are explored. (need tag for R^2 -> R^2 example, general)

• ##### Definition of linear transformation, coordinate vector space math.la.d.lintrans.coord
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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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# 2-d linear transformations: dilation, projection, and shear

Examples of special types of linear transformation from R^2 to R^2: dilation, projection, and shear. (Some issues with the video: things re-start around the 10 second mark, and at 3:46 the word "projection" is said, when it should be "transformation". Also, at the end maybe it could be described why it is called a 'shear'.)

• ##### Example of a linear transformation on R^2: shear math.la.e.lintrans.shear.r2
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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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# 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 from R^n to R^m

Motivation of the definition of a linear transformation using properties of matrices; examples; geometric intuition; matrix representation of a linear transformation

• ##### Example of a linear transformation on R^2: shear math.la.e.lintrans.shear.r2
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September 3rd, 2017
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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
GFDL-1.2
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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
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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
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# 2-d linear transformations: dilation, projection, and shear

Examples of special types of linear transformation from R^2 to R^2: dilation, projection, and shear. (Some issues with the video: things re-start around the 10 second mark, and at 3:46 the word "projection" is said, when it should be "transformation". Also, at the end maybe it could be described why it is called a 'shear'.)

• ##### Example of a linear transformation on R^2: shear math.la.e.lintrans.shear.r2
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February 15th, 2017
6 years ago
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# Linear Transformation Applet

Visualize 2-d linear transformations by looking at the image of geometric object. (Need topic: Visualize a linear transformation on R^2 by its effect on a region.)

• ##### Geometric properties of linear transformations on R^2 math.la.c.lintrans.geometric.r2
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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
Created On
June 8th, 2017
5 years ago
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# In-class activity: Linear transformations

In-class activity to be completed after an introduction to transformations and ideally in teams. In part 1, students are guided to discover the theorem describing the matrix of a linear transformation from R^n to R^m. In part 2, students learn the one-to-one and onto properties of linear transformations, and are asked to relate these properties to the properties of the matrices (linear independence of columns and columns spanning the codomain).

• ##### Definition of the standard matrix for a linear transformation, coordinate setting math.la.d.lintrans.mat.basis.standard.coord
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5 years ago
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# Linear transformations from R^n to R^m

Motivation of the definition of a linear transformation using properties of matrices; examples; geometric intuition; matrix representation of a linear transformation

• ##### Example of a linear transformation on R^2: shear math.la.e.lintrans.shear.r2
Created On
September 3rd, 2017
5 years ago
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3
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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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5 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)
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6 years ago
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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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# 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 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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# 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
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# Is AB = BA for matrices?

A 2x2 example is used to show that AB does not always equal BA.

• ##### Example of multiplying 2x2 matrices math.la.e.mat.mult.2x2
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February 17th, 2017
6 years ago
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# Is AB = BA for matrices? Example 2

Example 3x2 and a 2x3 matrices are used to show that AB does not always equal BA

• ##### Example of multiplying nonsquare matrices math.la.e.mat.mult.nonsquare
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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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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
Created On
February 17th, 2017
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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September 11th, 2017
5 years ago
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3
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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.

• ##### Matrix multiplication is associative. math.la.t.mat.mult.assoc
GFDL-1.2
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September 11th, 2017
5 years ago
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3
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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.

• ##### The transpose of a product of matrices is the product of the transposes in reverse order. math.la.t.mat.mult.transpose
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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.

• ##### Matrix multiplication is distributive over matrix addition. math.la.t.mat.mult.distributive
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September 11th, 2017
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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.

• ##### The identity matrix is the identity for matrix multiplication. math.la.t.mat.mult.identity
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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
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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
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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
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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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# 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
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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
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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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# Invertible Matrix Theorem

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

• ##### Definition of matrix inverse math.la.d.mat.inv
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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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# 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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# 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
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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
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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
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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.

• ##### Matrix transpose commutes with matrix inverse. math.la.t.mat.inv.transpose
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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 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
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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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# 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
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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.

• ##### 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
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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 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
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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
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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
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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 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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# 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
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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
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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
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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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# 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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# 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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# 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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# 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
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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 - 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
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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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# 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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# 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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# 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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# 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
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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
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5 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.

• ##### If A and B are n-by-n matrices, then det(AB)=det(A)det(B). math.la.t.mat.det.product
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September 11th, 2017
5 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
GFDL-1.2
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5 years ago
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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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January 1st, 2017
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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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December 28th, 2016
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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# 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 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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# 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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# 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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# 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.

• ##### 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.

• ##### The zero scalar multiplied by any vector equals the zero vector. math.la.t.vsp.scalar.mult.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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# 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.function
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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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# 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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# 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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# 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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# 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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# 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.

• ##### Definition of linear transformation, arbitrary vector space math.la.d.lintrans.arb
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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
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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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# 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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# 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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# 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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# 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
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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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# 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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# 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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# 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 $$\complex{m}\text{.}$$

• ##### 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
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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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# 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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# 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
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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
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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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# 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
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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
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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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# 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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# 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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# 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
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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
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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
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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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# 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
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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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# 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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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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# 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
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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
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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
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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
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# Scalar Equation of a Plane and Projections Lesson

This is from the University of Waterloo. It includes content about Projections, as well as some content from the Multivariable Calculus. These notions are developed in Euclidean Space.

• ##### Definition of (orthogonal) projection of one vector onto another vector math.la.d.vec.projection
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October 23rd, 2013
9 years ago
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# Projections Quiz

This is a quiz from the University of Waterloo. It is a quiz about projections that is strictly in R^n. It additionally asks questions on perpendicular vectors and cross products.

• ##### Formula for the (orthogonal) projection of one vector onto another vector math.la.t.vec.projection
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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
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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 $$\complex{m}\text{.}$$

• ##### An orthogonal set of nonzero vectors is linearly independent. math.la.t.vec.orthogonal_set.linindep
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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
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5 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 orthogonal set of vectors math.la.d.vec.orthogonal_set
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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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# Orthogonal projection

Orthogonal projection onto subspace in R^n minimizes distance; projection formula simplification for orthonormal bases; relation to orthogonal matrices

• ##### Definition of orthogonal projection onto a subspace math.la.d.vec.projection.subspace
Created On
August 21st, 2017
5 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 $$\complex{m}\text{.}$$

• ##### The Gram-Schmidt process converts a basis into an orthogonal basis. math.la.t.gramschmidt
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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
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August 21st, 2017
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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
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