Solving Systems of Linear Equations - A First Course in Linear Algebra

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

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

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

• Definition of solution set of a system of linear equations math.la.d.linsys.soln_set
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What is Linear Algebra? - A First Course in Linear Algebra

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

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

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

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

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

• Equation operations on a linear system give an equivalent system. math.la.t.linsys.op
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September 11th, 2017
6 years ago
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2
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Textbook
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Solving Systems of Linear Equations - A First Course in Linear Algebra

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

• Definition of equation operations on a linear system math.la.d.linsys.op
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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Textbook
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English
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Types of Solution Sets - A First Course in Linear Algebra

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

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

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

• Definition of inconsistent linear system math.la.d.linsys.inconsistent
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6 years ago
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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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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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Vector Arithmetic

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

• math.la.t.vec.sum.geometric.RnCn
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University of Waterloo Math Online -

• Definition of linear combination of vectors, coordinate vector space math.la.d.vec.lincomb.coord
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Slides for the accompanying video from University of Waterloo.

• Definition of scalar, coordinate vector space math.la.d.scalar
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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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9 years ago
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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
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August 25th, 2017
6 years ago
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Reduced Row-Echelon Form - A First Course in Linear Algebra

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

• math.la.c.mat.entry
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September 11th, 2017
6 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.

• 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
6 years ago
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Reduced Row-Echelon Form - A First Course in Linear Algebra

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

• Definition of solution vector of a linear system math.la.d.vec.solution
GFDL-1.2
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6 years ago
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Types of Solution Sets - A First Course in Linear Algebra

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

• Definition of free/independent variable in a linear system math.la.d.linsys.variable.independent
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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Reduced Row-Echelon Form - A First Course in Linear Algebra

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

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

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

• Definition of matrix representation of a linear system math.la.d.linsys.mat.repn
GFDL-1.2
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6 years ago
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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
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6 years ago
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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
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6 years ago
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Reduced Row-Echelon Form - A First Course in Linear Algebra

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

• Definition of leading entry in a row of a matrix math.la.d.mat.row.leading
GFDL-1.2
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6 years ago
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3
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Matrix Operations - A First Course in Linear Algebra

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

• math.la.d.mat.m_by_n.set
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When does a linear system have a unique solution?

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

• Definition of consistent linear system math.la.d.linsys.consistent
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6 years ago
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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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6 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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August 22nd, 2017
6 years ago
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Solving homogeneous systems of equations

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

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

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

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

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

• Homogeneous linear systems are consistent. math.la.d.linsys.homog.consistent
GFDL-1.2
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6 years ago
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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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Linear Combinations - A First Course in Linear Algebra

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

• The solutions to a nonhomogeneous system are given by a particular solution plus the solutions to the homogeneous system. math.la.t.linsys.nonhomog.particular_plus_homog
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II Linear Geometry

II.1 Vectors in Space

University of Waterloo Math Online -

• Definition of linear combination of vectors, coordinate vector space math.la.d.vec.lincomb.coord
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October 23rd, 2013
9 years ago
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Dot Product and Cross Product Lesson

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

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

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

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

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

• Two vectors are orthogonal if and only if the Pythagorean Theorem holds. math.la.t.vec.orthogonal
Created On
August 22nd, 2017
6 years ago
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2
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Video
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Orthogonality - A First Course in Linear Algebra

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

• Definition of 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 $$\complex{m}\text{.}$$

• math.la.d.vec.orthogonal.coord
GFDL-1.2
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September 11th, 2017
6 years ago
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When does a linear system have a unique solution?

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

• Definition of consistent linear system math.la.d.linsys.consistent
Created On
February 15th, 2017
6 years ago
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3
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Video
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Example
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Using matrices to solve linear systems

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

• Example of putting a matrix in echelon form math.la.e.mat.echelon.of
Created On
August 21st, 2017
6 years ago
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2
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Video
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English
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Echelon form

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

• Example of putting a matrix in echelon form math.la.e.mat.echelon.of
Created On
August 25th, 2017
6 years ago
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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.

• Equation operations on a linear system give an equivalent system. math.la.t.linsys.op
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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2
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Textbook
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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
Submitted At
September 11th, 2017
6 years ago
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2
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Reduced Row-Echelon Form - A First Course in Linear Algebra

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

• Definition of leading entry in a row of a matrix math.la.d.mat.row.leading
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
Type
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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
Submitted At
September 11th, 2017
6 years ago
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3
Type
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
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September 11th, 2017
6 years ago
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2
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Vector space Part 1: what makes R^n a "space"

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

• Axioms of a vector space, arbitrary vector space math.la.d.vsp.axioms.arb
CC-BY-SA-4.0
Created On
January 1st, 2017
6 years ago
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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
CC-BY-SA-4.0
Created On
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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January 1st, 2017
6 years ago
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Vector spaces

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

• The negative of a vector equals the vector multiplied by -1. math.la.t.vsp.vector.negative
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August 25th, 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.

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

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

• The zero vector multiplied by any scalar equals the zero vector. math.la.t.vsp.vector.mult.z
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September 11th, 2017
6 years ago
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Vector Spaces - A First Course in Linear Algebra

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

• math.la.e.vsp.mat.m_by_n
GFDL-1.2
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Subspaces Part 1: definition of a subspace

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

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

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

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

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

• Definition of zero subspace, arbitrary vector space math.la.d.vsp.subspace.z
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September 3rd, 2017
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Subspaces - A First Course in Linear Algebra

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

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

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

• math.la.t.vec.span.subspace.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.

• Definition of spanning set for a subspace, arbitrary vector space math.la.d.vsp.span.set.arb
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Subspaces - A First Course in Linear Algebra

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

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

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

• math.la.t.vsp.subspace.lincomb.arb
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6 years ago
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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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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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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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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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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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6 years ago
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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 $$\complex{m}\text{.}$$

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

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

• 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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Definition and properties of matrix transpose

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

• Matrix transpose is an involution. math.la.t.mat.transpose.involution
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February 17th, 2017
6 years ago
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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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June 9th, 2017
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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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Matrix equations and spanning sets

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

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

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

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

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

• Equivalence theorem: the equation Ax=b has a solution for all b. math.la.t.equiv.mat.eqn.unique
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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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Matrix Operations - A First Course in Linear Algebra

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

• Definition of transpose of a matrix math.la.d.mat.transpose
GFDL-1.2
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September 11th, 2017
6 years ago
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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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6 years ago
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Column and Row Spaces - A First Course in Linear Algebra

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

• Row equivalent matrices have the same row space. math.la.d.mat.row_space.row_equiv
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6 years ago
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Dimension - A First Course in Linear Algebra

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

• Definition of rank of a matrix math.la.d.mat.rank
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6 years ago
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2
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Column and Row Spaces - A First Course in Linear Algebra

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

• Definition of column space of a matrix math.la.d.mat.col_space
GFDL-1.2
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September 11th, 2017
6 years ago
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Dimension - A First Course in Linear Algebra

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

• If A is a matrix, then the rank of A plus the nullity of A equals the number of columns of A. math.la.t.mat.ranknullity
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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Properties of Dimension - A First Course in Linear Algebra

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

• The row space and the column space of a matrix have the same dimension. math.la.t.mat.row_space.col_space
GFDL-1.2
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6 years ago
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Column and Row Spaces - A First Course in Linear Algebra

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

• Definition of row space of a matrix math.la.d.mat.row_space
GFDL-1.2
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6 years ago
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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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6 years ago
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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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Linear Transformations - A First Course in Linear Algebra

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

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

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

• Definition of isomorphism between vector spaces math.la.d.vsp.isomorphism
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Vector Representations - A First Course in Linear Algebra

You may have noticed that many questions about elements of abstract vector spaces eventually become questions about column vectors or systems of equations. Example SM32 would be an example of this. We will make this vague idea more precise in this section.

• math.la.t.vsp.dim.isomorphic
GFDL-1.2
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6 years ago
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Invertible Linear Transformations - A First Course in Linear Algebra

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

• math.la.t.vsp.isomorphic.dim
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Vector Representations - A First Course in Linear Algebra

You may have noticed that many questions about elements of abstract vector spaces eventually become questions about column vectors or systems of equations. Example SM32 would be an example of this. We will make this vague idea more precise in this section.

• math.la.t.vsp.isomorphic.rncn
GFDL-1.2
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September 11th, 2017
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Why an abstract linear transformation maps 0 to 0

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

• Definition of linear transformation, arbitrary vector space math.la.d.lintrans.arb
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February 15th, 2017
6 years ago
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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
Created On
June 8th, 2017
6 years ago
Views
2
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Handout
Perspective
Application
Language
English
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text/html; charset=utf-8

Column space of a matrix

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

• Definition of range of linear transformation, arbitrary vector space math.la.d.lintrans.range.arb
Created On
August 25th, 2017
6 years ago
Views
3
Type
Video
Language
English
Content Type
text/html; charset=utf-8

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
Submitted At
September 11th, 2017
6 years ago
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2
Type
Textbook
Language
English
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text/html

Linear Transformations - A First Course in Linear Algebra

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

• A linear transformation of a linear combination is the linear combination of the linear transformation math.la.t.lintrans.lincomb
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

Linear Transformations - A First Course in Linear Algebra

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

• Definition of linear transformation, arbitrary vector space math.la.d.lintrans.arb
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

Linear Transformations - A First Course in Linear Algebra

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

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

Linear Transformations - A First Course in Linear Algebra

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

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

Column space of a matrix

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

• Definition of range of linear transformation, arbitrary vector space math.la.d.lintrans.range.arb
Created On
August 25th, 2017
6 years ago
Views
3
Type
Video
Language
English
Content Type
text/html; charset=utf-8

Injective Linear Transformations - A First Course in Linear Algebra

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

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

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

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

Invertible Linear Transformations - A First Course in Linear Algebra

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

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

Invertible Linear Transformations - A First Course in Linear Algebra

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

• Definition of rank of a linear transformation math.la.d.lintrans.rank
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

Injective Linear Transformations - A First Course in Linear Algebra

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

• Definition of kernel of linear transformation, arbitrary vector space math.la.d.lintrans.kernel.arb
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

Invertible Linear Transformations - A First Course in Linear Algebra

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

• Definition of nullity of a linear transformation math.la.d.lintrans.nullity
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

Injective Linear Transformations - A First Course in Linear Algebra

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

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

Matrix equations as motivation for basis and span

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

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

How to multiply matrices

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

• Definition of matrix-vector product, as a linear combination of column vectors math.la.d.mat.vec.prod
Created On
February 17th, 2017
6 years ago
Views
2
Type
Video
Timeframe
Review
Language
English
Content Type
text/html; charset=utf-8

Matrix Multiplication - A First Course in Linear Algebra

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

• Definition of matrix-vector product, as a linear combination of column vectors math.la.d.mat.vec.prod
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

Matrix Representations - A First Course in Linear Algebra

We have seen that linear transformations whose domain and codomain are vector spaces of columns vectors have a close relationship with matrices (Theorem MBLT, Theorem MLTCV). In this section, we will extend the relationship between matrices and linear transformations to the setting of linear transformations between abstract vector spaces.

• Definition of matrix representation of a linear transformation with respect to bases of the spaces, arbitrary vector space math.la.d.lintrans.mat.repn.arb
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
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text/html

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
Created On
August 25th, 2017
6 years ago
Views
2
Type
Video
Language
English
Content Type
text/html; charset=utf-8

Linear Transformations - A First Course in Linear Algebra

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

• Matrices act as a transformations by multiplying vectors math.la.c.transformation.matrix
GFDL-1.2
Submitted At
September 11th, 2017
6 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
Perspective
Introduction
Language
English
Content Type
text/html; charset=utf-8

Notation for matrix entries, size of a matrix, etc

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

• Definition of zero matrix math.la.d.mat.zero
Created On
February 17th, 2017
6 years ago
Views
3
Type
Video
Timeframe
Pre-class
Perspective
Introduction
Language
English
Content Type
text/html; charset=utf-8

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
Created On
February 17th, 2017
6 years ago
Views
3
Type
Video
Timeframe
Pre-class
Perspective
Introduction
Language
English
Content Type
text/html; charset=utf-8

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
Views
2
Type
Handout
Timeframe
In-class
Language
English
Content Type
text/html; charset=utf-8

Matrix Operations - A First Course in Linear Algebra

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

• Definition of matrix-scalar multiplication math.la.d.mat.scalar.mult
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

Matrix Operations - A First Course in Linear Algebra

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

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

Matrix Representations - A First Course in Linear Algebra

We have seen that linear transformations whose domain and codomain are vector spaces of columns vectors have a close relationship with matrices (Theorem MBLT, Theorem MLTCV). In this section, we will extend the relationship between matrices and linear transformations to the setting of linear transformations between abstract vector spaces.

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

Matrix Representations - A First Course in Linear Algebra

We have seen that linear transformations whose domain and codomain are vector spaces of columns vectors have a close relationship with matrices (Theorem MBLT, Theorem MLTCV). In this section, we will extend the relationship between matrices and linear transformations to the setting of linear transformations between abstract vector spaces.

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

Matrix operations

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

• The transpose of a product of matrices is the product of the transposes in reverse order. math.la.t.mat.mult.transpose
Created On
August 25th, 2017
6 years ago
Views
2
Type
Video
Language
English
Content Type
text/html; charset=utf-8

Matrix Multiplication - A First Course in Linear Algebra

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

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

Matrix Multiplication - A First Course in Linear Algebra

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

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

Matrix Representations - A First Course in Linear Algebra

We have seen that linear transformations whose domain and codomain are vector spaces of columns vectors have a close relationship with matrices (Theorem MBLT, Theorem MLTCV). In this section, we will extend the relationship between matrices and linear transformations to the setting of linear transformations between abstract vector spaces.

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

Linear Transformations - A First Course in Linear Algebra

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

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

The Matrix Inverse - Definition

Motivation and definition of the inverse of a matrix

• Matrix inverses are unique: if A and B are square matrices, then AB=I implies that A=B^-1 and B=A^-1. math.la.t.mat.inv.unique
(CC-BY-NC-SA-4.0 OR CC-BY-SA-4.0)
Created On
January 5th, 2017
6 years ago
Views
3
Type
Video
Timeframe
Pre-class
Perspective
Introduction
Language
English
Content Type
text/html; charset=utf-8

Notation for matrix entries, size of a matrix, etc

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

• Definition of zero matrix math.la.d.mat.zero
Created On
February 17th, 2017
6 years ago
Views
3
Type
Video
Timeframe
Pre-class
Perspective
Introduction
Language
English
Content Type
text/html; charset=utf-8

How to multiply matrices

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

• Definition of matrix-vector product, as a linear combination of column vectors math.la.d.mat.vec.prod
Created On
February 17th, 2017
6 years ago
Views
2
Type
Video
Timeframe
Review
Language
English
Content Type
text/html; charset=utf-8

Matrix Inverse

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

• The identity matrix is the identity for matrix multiplication. math.la.t.mat.mult.identity
Created On
February 17th, 2017
6 years ago
Views
2
Type
Video
Timeframe
Pre-class
Perspective
Introduction
Language
English
Content Type
text/html; charset=utf-8

Matrix inverses

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

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

Nonsingular Matrices - A First Course in Linear Algebra

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

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

Matrix Operations - A First Course in Linear Algebra

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

• Definition of the diagonal of a matrix math.la.d.mat.thediagonal
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

Matrix Multiplication - A First Course in Linear Algebra

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

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

Similarity and Diagonalization - A First Course in Linear Algebra

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

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

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
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
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text/html

Orthonormal Diagonalization - A First Course in Linear Algebra

We have seen in Section SD that under the right conditions a square matrix is similar to a diagonal matrix. We recognize now, via Theorem SCB, that a similarity transformation is a change of basis on a matrix representation. So we can now discuss the choice of a basis used to build a matrix representation, and decide if some bases are better than others for this purpose. This will be the tone of this section. We will also see that every matrix has a reasonably useful matrix representation, and we will discover a new class of diagonalizable linear transformations. First we need some basic facts about triangular matrices.

• Definition of an upper triangular matrix math.la.d.mat.triangular.upper
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

Orthonormal Diagonalization - A First Course in Linear Algebra

We have seen in Section SD that under the right conditions a square matrix is similar to a diagonal matrix. We recognize now, via Theorem SCB, that a similarity transformation is a change of basis on a matrix representation. So we can now discuss the choice of a basis used to build a matrix representation, and decide if some bases are better than others for this purpose. This will be the tone of this section. We will also see that every matrix has a reasonably useful matrix representation, and we will discover a new class of diagonalizable linear transformations. First we need some basic facts about triangular matrices.

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

The Matrix Inverse - Definition

Motivation and definition of the inverse of a matrix

• Matrix inverses are unique: if A and B are square matrices, then AB=I implies that A=B^-1 and B=A^-1. math.la.t.mat.inv.unique
(CC-BY-NC-SA-4.0 OR CC-BY-SA-4.0)
Created On
January 5th, 2017
6 years ago
Views
3
Type
Video
Timeframe
Pre-class
Perspective
Introduction
Language
English
Content Type
text/html; charset=utf-8

Matrix Inverse

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

• The identity matrix is the identity for matrix multiplication. math.la.t.mat.mult.identity
Created On
February 17th, 2017
6 years ago
Views
2
Type
Video
Timeframe
Pre-class
Perspective
Introduction
Language
English
Content Type
text/html; charset=utf-8

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

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

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

Inverse of a Matrix: In-Class Activities

Suggested classroom activities on matrix inverses.

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

Lagrange Interpolation - Ximera

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

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

Invertible Matrix Theorem

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

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

Matrix Inverses, definition and examples

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

• The inverse of a matrix can be used to solve a linear system. math.la.t.eqn.mat.inv
Created On
August 22nd, 2017
6 years ago
Views
4
Type
Video
Language
English
Content Type
text/html; charset=utf-8

Matrix inverses

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

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

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

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

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

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

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

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

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

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

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

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

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

• 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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September 11th, 2017
6 years ago
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Matrix Inverses and Nonsingular Matrices - A First Course in Linear Algebra

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

• Equivalence theorem: the matrix A has an inverse. math.la.t.equiv.inv
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Lagrange Interpolation - Ximera

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

• Definition of linear transformation, arbitrary vector space math.la.d.lintrans.arb
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June 8th, 2017
6 years ago
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Application
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Change of Basis - A First Course in Linear Algebra

We have seen in Section MR that a linear transformation can be represented by a matrix, once we pick bases for the domain and codomain. How does the matrix representation change if we choose different bases? Which bases lead to especially nice representations? From the infinite possibilities, what is the best possible representation? This section will begin to answer these questions. But first we need to define eigenvalues for linear transformations and the change-of-basis matrix.

• math.la.t.vsp.change_of_basis
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September 11th, 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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September 11th, 2017
6 years ago
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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
Created On
October 23rd, 2013
9 years ago
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Video
Perspective
Introduction
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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
Created On
October 23rd, 2013
9 years ago
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2
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Unknown
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Post-class
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Example
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English
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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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August 25th, 2017
6 years ago
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Orthogonality - A First Course in Linear Algebra

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

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

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

• The Gram-Schmidt process converts a basis into an orthogonal basis. math.la.t.gramschmidt
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September 11th, 2017
6 years ago
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Orthogonality - A First Course in Linear Algebra

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

• Definition of orthogonal set of vectors math.la.d.vec.orthogonal_set
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September 11th, 2017
6 years ago
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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
6 years ago
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4
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Video
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English
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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
Created On
February 17th, 2017
6 years ago
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2
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Video
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Pre-class
Perspective
Introduction
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English
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Matrix Inverses, definition and examples

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

• The inverse of a matrix can be used to solve a linear system. math.la.t.eqn.mat.inv
Created On
August 22nd, 2017
6 years ago
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4
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Video
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English
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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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September 11th, 2017
6 years ago
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Determinants and row operations

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

• The determinant of a triangular matrix is the product of the entries on the diagonal. math.la.t.mat.det.trianglar
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August 22nd, 2017
6 years ago
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4
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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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September 11th, 2017
6 years ago
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Properties of Determinants of Matrices - A First Course in Linear Algebra

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

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

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

• math.la.t.mat.row.equal
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
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Textbook
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English
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Determinants and their relation to column operations and products

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

• Theorem describing the effect of elementary row operations on the determinant of a matrix. math.la.t.mat.det.elementaryoperations
Created On
August 25th, 2017
6 years ago
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3
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Video
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English
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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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September 11th, 2017
6 years ago
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3
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Determinants and their relation to column operations and products

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

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

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

• If A and B are n-by-n matrices, then det(AB)=det(A)det(B). math.la.t.mat.det.product
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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Textbook
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English
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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
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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Textbook
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English
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Determinant of a Matrix - A First Course in Linear Algebra

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

• Definition of cofactor of a matrix math.la.d.mat.cofactor
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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Similarity of matrices

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

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

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

• Definition of similar matrices math.la.d.mat.similar
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September 11th, 2017
6 years ago
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Diagonalization of a matrix

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

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

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

• Definition of matrix diagonalization math.la.d.mat.diagonalization
GFDL-1.2
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September 11th, 2017
6 years ago
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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
Created On
June 9th, 2017
6 years ago
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2
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Handout
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Pre-class
Perspective
Introduction
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English
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Geometry of eigenvectors

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

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

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

• An n-by-n matrix with n distinct eigenvalues is diagonalizable. math.la.t.mat.diagonalizable.distinct
Created On
August 25th, 2017
6 years ago
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3
Type
Video
Language
English
Content Type
text/html; charset=utf-8

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
Created On
September 3rd, 2017
6 years ago
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2
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Video
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English
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Similarity and Diagonalization - A First Course in Linear Algebra

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

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

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

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

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

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

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

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

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

• Definition of eigenvalue(s) of a matrix math.la.d.mat.eig
GFDL-1.2
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September 11th, 2017
6 years ago
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3
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Change of Basis - A First Course in Linear Algebra

We have seen in Section MR that a linear transformation can be represented by a matrix, once we pick bases for the domain and codomain. How does the matrix representation change if we choose different bases? Which bases lead to especially nice representations? From the infinite possibilities, what is the best possible representation? This section will begin to answer these questions. But first we need to define eigenvalues for linear transformations and the change-of-basis matrix.

• math.la.d.lintrans.eigvec
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
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Textbook
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Properties of Eigenvalues and Eigenvectors - A First Course in Linear Algebra

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

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

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

• math.la.d.mat.eig.multiplicity.algebraic
GFDL-1.2
Submitted At
September 11th, 2017
6 years ago
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3
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Textbook
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English
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