resources

# The Matrix Inverse - Definition

Motivation and definition of the inverse of a matrix

##### math.la.t.mat.inv.unique
(CC-BY-NC-SA-4.0 OR CC-BY-SA-4.0)
Created On
January 5th, 2017
6 years ago
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3
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# 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.

##### math.la.t.mat.eqn.linsys
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February 15th, 2017
6 years ago
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# Matrix equations as motivation for basis and span

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

##### math.la.t.mat.eqn.lincomb
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February 15th, 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.

##### math.la.d.linsys.consistent
Created On
February 15th, 2017
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.

##### math.la.d.mat.eqn
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February 15th, 2017
6 years ago
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# Inconsistent linear systems

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

##### math.la.e.linsys.3x3.soln.row_reduce.z
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February 15th, 2017
6 years ago
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4
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# CEMC Courseware - The Intersection of Three Planes

Applet to show the configuration of planes corresponding to a 3x3 system having 0, 1, or infinitely many solutions. (need to make tags for this topic)

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February 15th, 2017
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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February 15th, 2017
6 years ago
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# How to find out if a set of vectors is linearly independent?

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

##### math.la.d.vec.linindep.coord
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February 15th, 2017
6 years ago
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# Definition of Linear Transformation

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

##### math.la.d.lintrans.coord
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February 15th, 2017
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.

##### math.la.d.lintrans.arb
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February 15th, 2017
6 years ago
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2
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# 2-d linear transformations: dilation, projection, and shear

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

##### math.la.e.lintrans.shear.r2
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February 15th, 2017
6 years ago
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# Solving Ax=0

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

##### math.la.e.mat.eqn.3x3.homog.solve
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February 15th, 2017
6 years ago
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3
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# Linear Transformation Applet

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

##### math.la.c.lintrans.geometric.r2
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February 15th, 2017
6 years ago
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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.

##### math.la.d.mat.zero
Created On
February 17th, 2017
6 years ago
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3
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# Matrix Operations: Sums Scalar Multiplication

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

##### math.la.t.mat.scalar.mult.commut_assoc
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February 17th, 2017
6 years ago
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3
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# How to multiply matrices

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

##### math.la.d.mat.vec.prod
Created On
February 17th, 2017
6 years ago
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2
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# Is AB = BA for matrices?

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

##### math.la.e.mat.mult.2x2
Created On
February 17th, 2017
6 years ago
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3
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# Is AB = BA for matrices? Example 2

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

##### math.la.e.mat.mult.nonsquare
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February 17th, 2017
6 years ago
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3
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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.

##### math.la.t.mat.transpose.involution
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February 17th, 2017
6 years ago
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2
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# In-class activities for matrix operations

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

##### math.la.c.mat.mult.cancellation
Created On
February 17th, 2017
6 years ago
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2
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English
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# Matrix Inverse

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

##### math.la.t.mat.mult.identity
Created On
February 17th, 2017
6 years ago
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2
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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)

##### math.la.t.mat.det.2x2
Created On
February 17th, 2017
6 years ago
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2
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# Finding the inverse of a matrix and using it to solve a linear system

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

##### math.la.t.eqn.mat.inv
Created On
February 19th, 2017
6 years ago
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3
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# Inverse of a Matrix: In-Class Activities

Suggested classroom activities on matrix inverses.

##### math.la.t.mat.inv.augmented
Created On
February 19th, 2017
6 years ago
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2
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# Equivalence theorem for nonsingular matrices

Eleven different statements are shown to be equivalent to "The matrix A is nonsingular". (need a tag for a theorem that contains several of these equivalences)

Created On
February 19th, 2017
6 years ago
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3
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English
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# Invertible Matrix Theorem: In-Class Activities

A guide for discussing the many statements equivalent to "The matrix A is nonsingular". (need a tag for multiple equivalences)

Created On
February 19th, 2017
6 years ago
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3
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In-class
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English
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# Vector Arithmetic

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

##### math.la.t.vec.sum.geometric.RnCn
Created On
February 19th, 2017
6 years ago
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2
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# Linear combinations of vectors

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

##### math.la.t.mat.eqn.lincomb
Created On
February 20th, 2017
6 years ago
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2
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# Span of a set of vectors

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

##### math.la.c.vec.span.geometric.rncn
Created On
February 20th, 2017
6 years ago
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3
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# Vector Equations: In-Class Activities

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

##### math.la.t.linsys.vec
Created On
February 20th, 2017
6 years ago
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2
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English
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# Row reducing a 4x4 matrix

A 4x4 matrix is put into row echelon form (not RREF), with motivation and all the details (need a 4x4 ref tag)

Created On
February 20th, 2017
6 years ago
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2
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# RREF In-Class Activities

Suggestions for in-class activities for row reduction. (Need to add tags. The refrenced videos should be separate assets.)

Created On
February 20th, 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.

##### math.la.d.vsp.axioms.arb
CC-BY-SA-4.0
Created On
January 1st, 2017
6 years ago
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3
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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...

##### math.la.d.lintrans.arb
Created On
May 25th, 2017
6 years ago
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2
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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...

##### math.la.d.vsp.axioms.arb
CC-BY-SA-4.0
Created On
December 28th, 2016
6 years ago
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3
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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.

##### math.la.d.vsp.axioms.arb
CC-BY-SA-4.0
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January 1st, 2017
6 years ago
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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.

##### math.la.d.vsp.subspace.arb
CC-BY-SA-4.0
Created On
January 3rd, 2017
6 years ago
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3
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# Linear Combinations and Span

In-class activity for linear combinations and span.

##### math.la.d.vec.lincomb.coord
GFDL-1.3
Created On
June 8th, 2017
6 years ago
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3
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# Linear Independence in-class activity

Linear independence in-class activity

##### math.la.d.vec.lindep.relation
GFDL-1.3
Created On
June 8th, 2017
6 years ago
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3
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# Pre-class quiz on Linear Transformations

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

##### math.la.t.lintrans.mat.basis.standard.coord
Created On
June 8th, 2017
6 years ago
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2
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# In-class activity: Linear transformations

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

##### math.la.d.lintrans.mat.basis.standard.coord
Created On
June 8th, 2017
6 years ago
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2
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In-class
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English
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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.

##### math.la.t.mat.ranknullity
Created On
June 9th, 2017
6 years ago
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2
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In-class
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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.

##### math.la.d.mat.eig
Created On
June 9th, 2017
6 years ago
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2
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# Lagrange Interpolation

This is an activity on how linear algebra can be used to find the Lagrange polynomials to construct the polynomial of degree n that passes through n+1 distinct points. This activity requires knowledge of vector spaces (particularly polynomial spaces), linear transformations, and matrix inverses. It would be helpful to have a tag for applications of linear algebra for material like this.

CC-BY-NC-SA-4.0
Created On
June 9th, 2017
6 years ago
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2
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# Vector Addition and Scalar Multiplication

University of Waterloo Math Online -

##### math.la.d.vec.lincomb.coord
Created On
October 23rd, 2013
9 years ago
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# Vector Addition and Scalar Multiplication

Slides for the accompanying video from University of Waterloo.

##### math.la.d.scalar
Created On
October 23rd, 2013
9 years ago
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3
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# Properties of Vectors and Spanning

From the University of Waterloo Math Online

##### math.la.c.vec.span.geometric.rncn
Created On
October 23rd, 2013
9 years ago
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# Properties of Vectors and Spanning

Slides from the corresponding video from the University of Waterloo.

##### math.la.c.vec.span.geometric.rncn
Created On
October 23rd, 2013
9 years ago
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3
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# University of Waterloo Math Online - Lesson: Linear Independence and Surfaces

Video Lesson from University of Waterloo.

##### math.la.t.vec.lindep.coord
Created On
October 23rd, 2013
9 years ago
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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.

##### math.la.d.lintrans.arb
Created On
June 8th, 2017
6 years ago
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2
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# Vector Addition and Scalar Multiplication Quiz

Quiz from the University of Waterloo.

##### math.la.t.vec.lindep.coord
Created On
October 23rd, 2013
9 years ago
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4
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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.

##### math.la.d.crossproduct
Created On
October 23rd, 2013
9 years ago
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3
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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.

##### math.la.d.vec.orthogonal
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October 23rd, 2013
9 years ago
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3
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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.

##### math.la.d.vec.projection
Created On
October 23rd, 2013
9 years ago
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3
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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.

##### math.la.t.vec.projection
Created On
October 23rd, 2013
9 years ago
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2
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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.

##### math.la.d.vsp.subspace.arb
CC-BY-SA-4.0
Created On
June 9th, 2017
6 years ago
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3
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# Linear transformations: injective and surjective properties

One-to-one and onto properties of liner transformations: definition, geometric interpretation, examples, connection with properties of the columns of a matrix representation.

##### math.la.c.lintrans.geometric
Created On
August 21st, 2017
5 years ago
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3
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# Using matrices to solve linear systems

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

##### math.la.e.mat.echelon.of
Created On
August 21st, 2017
5 years ago
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2
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# Invertible Matrix Theorem

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

##### math.la.d.mat.inv
Created On
August 21st, 2017
5 years ago
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3
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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

##### math.la.d.vec.projection.subspace
Created On
August 21st, 2017
5 years ago
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4
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# Diagonalization of Real Symmetric Matrices

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

##### math.la.t.mat.symmetric.spectral
Created On
August 21st, 2017
5 years ago
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4
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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.

##### math.la.t.vec.orthogonal
Created On
August 22nd, 2017
5 years ago
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2
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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.

##### math.la.t.eqn.mat.inv
Created On
August 22nd, 2017
5 years ago
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4
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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.

##### math.la.d.linsys.homog.nontrivial
Created On
August 22nd, 2017
5 years ago
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3
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# Determinants and row operations

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

##### math.la.t.mat.det.trianglar
Created On
August 22nd, 2017
5 years ago
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4
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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

##### math.la.d.mat.eigsp
Created On
August 25th, 2017
5 years ago
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2
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# Matrix inverses

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

##### math.la.t.equiv.identity
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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

##### math.la.t.mat.col_space.pivot
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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.

##### math.la.t.mat.diagonalizable.distinct
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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.

##### math.la.e.linsys.3x3.soln.homog.row_reduce.i
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# Bases for the nullspace and column space of a matrix

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

##### math.la.t.mat.col_space.pivot
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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

##### math.la.t.vsp.vector.negative
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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

##### math.la.d.vsp.basis.relative.coord
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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

##### math.la.e.mat.echelon.of
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# Linear transformations

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

##### math.la.e.lintrans.projection.r2
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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

##### math.la.t.mat.det.elementaryoperations
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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

##### 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.

##### math.la.t.mat.eqn.lincomb
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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

##### math.la.t.mat.col.orthonormal.inv.rn
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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

##### math.la.d.lintrans.range.arb
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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.

##### math.la.t.vsp.basis.coord.vector.arb
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# Linear transformations from R^n to R^m

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

##### math.la.e.lintrans.shear.r2
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# Characteristic equation of a matrix

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

##### math.la.t.mat.eig.triangular
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# Vectors and their linear combinations in R^n

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

##### math.la.e.vec.lincomb.weight.solve.r3
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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.

##### math.la.t.mat.similar.eig
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# Vector subspaces

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

##### math.la.d.vsp.subspace.z
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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.

##### math.la.d.linsys
GFDL-1.2
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# Solving Systems of Linear Equations - A First Course in Linear Algebra

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

##### math.la.d.linsys.soln_set
GFDL-1.2
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# What is Linear Algebra? - A First Course in Linear Algebra

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

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

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

##### math.la.d.linsys.soln
GFDL-1.2
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September 11th, 2017
5 years ago
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2
Type
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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.

##### math.la.d.linsys.equiv
GFDL-1.2
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September 11th, 2017
5 years ago
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3
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# Reduced Row-Echelon Form - A First Course in Linear Algebra

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

##### math.la.c.mat.entry
GFDL-1.2
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September 11th, 2017
5 years ago
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3
Type
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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.

##### math.la.t.linsys.op
GFDL-1.2
Submitted At
September 11th, 2017
5 years ago
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2
Type
Textbook
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English
Content 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.

##### math.la.d.vec.component.coord
GFDL-1.2
Submitted At
September 11th, 2017
5 years ago
Views
2
Type
Textbook
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English
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# Reduced Row-Echelon Form - A First Course in Linear Algebra

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

##### math.la.d.mat.constant
GFDL-1.2
Submitted At
September 11th, 2017
5 years ago
Views
3
Type
Textbook
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English
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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.

##### math.la.d.linsys.op
GFDL-1.2
Submitted At
September 11th, 2017
5 years ago
Views
3
Type
Textbook
Language
English
Content 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.

##### math.la.d.mat.row_op
GFDL-1.2
Submitted At
September 11th, 2017
5 years ago
Views
2
Type
Textbook
Language
English
Content 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.

##### math.la.d.mat.coeff
GFDL-1.2
Submitted At
September 11th, 2017
5 years ago
Views
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.

##### math.la.d.mat.augmented
GFDL-1.2
Submitted At
September 11th, 2017
5 years ago
Views
3
Type
Textbook
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English
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# Reduced Row-Echelon Form - A First Course in Linear Algebra

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

##### math.la.d.vec.solution
GFDL-1.2
Submitted At
September 11th, 2017
5 years ago
Views
3
Type
Textbook
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English
Content 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.

##### math.la.d.mat.row_equiv
GFDL-1.2
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September 11th, 2017
5 years ago
Views
2
Type
Textbook
Language
English
Content Type
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# Types of Solution Sets - A First Course in Linear Algebra

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

##### math.la.t.linsys.zoi
GFDL-1.2
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September 11th, 2017
5 years ago
Views
3
Type
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.

##### math.la.d.linsys.variable.independent
GFDL-1.2
Submitted At
September 11th, 2017
5 years ago
Views
3
Type
Textbook
Language
English
Content Type
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# Types of Solution Sets - A First Course in Linear Algebra

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

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

# Reduced Row-Echelon Form - A First Course in Linear Algebra

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

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

# 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.

##### math.la.d.linsys.homog
GFDL-1.2
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5 years ago
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2
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# Reduced Row-Echelon Form - A First Course in Linear Algebra

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

##### math.la.t.mat.rref.exists
GFDL-1.2
Submitted At
September 11th, 2017
5 years ago
Views
3
Type
Textbook
Language
English
Content 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.

##### math.la.d.linsys.mat.repn
GFDL-1.2
Submitted At
September 11th, 2017
5 years ago
Views
2
Type
Textbook
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English
Content 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.

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

# Types of Solution Sets - A First Course in Linear Algebra

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

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

# 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.

##### math.la.d.linsys.homog.consistent
GFDL-1.2
Submitted At
September 11th, 2017
5 years ago
Views
3
Type
Textbook
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English
Content Type
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# Homogeneous Systems of Equations - A First Course in Linear Algebra

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

##### math.la.t.linsys.homog.i
GFDL-1.2
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5 years ago
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3
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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.

##### math.la.d.mat.singular
GFDL-1.2
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5 years ago
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Type
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# Nonsingular Matrices - A First Course in Linear Algebra

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

##### math.la.t.equiv.mat.eqn.unique
GFDL-1.2
Submitted At
September 11th, 2017
5 years ago
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3
Type
Textbook
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English
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# Nonsingular Matrices - A First Course in Linear Algebra

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

##### math.la.t.equiv.identity
GFDL-1.2
Submitted At
September 11th, 2017
5 years ago
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2
Type
Textbook
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English
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# Nonsingular Matrices - A First Course in Linear Algebra

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

##### math.la.t.equiv.mat.eqn.unique.rep
GFDL-1.2
Submitted At
September 11th, 2017
5 years ago
Views
2
Type
Textbook
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English
Content Type
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# Homogeneous Systems of Equations - A First Course in Linear Algebra

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

##### math.la.d.mat.null_space.right
GFDL-1.2
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September 11th, 2017
5 years ago
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2
Type
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.

##### math.la.d.linsys.inconsistent
GFDL-1.2
Submitted At
September 11th, 2017
5 years ago
Views
3
Type
Textbook
Language
English
Content Type
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# Types of Solution Sets - A First Course in Linear Algebra

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

##### math.la.t.rref.pivot.free
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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.

##### math.la.d.mat.square
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# Vector Operations - A First Course in Linear Algebra

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

##### math.la.d.vec.sum.coord
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# Types of Solution Sets - A First Course in Linear Algebra

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

##### math.la.t.linsys.consistent.i
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# Vector Operations - A First Course in Linear Algebra

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

##### math.la.t.vec.sum.coord
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# Nonsingular Matrices - A First Course in Linear Algebra

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

##### math.la.d.mat.identity
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# Nonsingular Matrices - A First Course in Linear Algebra

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

##### math.la.t.equiv.nullspace
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# Vector Operations - A First Course in Linear Algebra

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

##### math.la.d.vec.rncn
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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.

##### math.la.t.linsys.nonhomog.particular_plus_homog
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# Orthogonality - A First Course in Linear Algebra

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

##### math.la.d.vec.conjugate.cn
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# Linear Independence - A First Course in Linear Algebra

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

##### math.la.d.vec.lindep.relation.trivial
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# Spanning Sets - A First Course in Linear Algebra

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

##### math.la.d.vec.span.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.t.vec.sum.conjugate.cn
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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).

##### math.la.t.vec.lindep.coord
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# Linear Independence - A First Course in Linear Algebra

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

##### math.la.t.equiv.col.linindep
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# Linear Independence - A First Course in Linear Algebra

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

##### math.la.d.vec.linindep.coord
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# Linear Independence - A First Course in Linear Algebra

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

##### math.la.t.vec.lindep.more.rncn
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# Orthogonality - A First Course in Linear Algebra

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

##### math.la.t.vec.scalar.mult.conjugate.cn
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# Vector Operations - A First Course in Linear Algebra

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

##### math.la.d.vec.scalar.mult.coord
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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.

##### math.la.d.linsys.homog.trivial
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# Orthogonality - A First Course in Linear Algebra

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

##### math.la.t.innerproduct.commutative.scalar.cn
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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.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.t.innerproduct.self.z.coord
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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.

##### math.la.t.linsys.soln.vector
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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.t.mat.row_equiv.linsys
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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).

##### math.la.t.vsp.span.basis
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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.transpose
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# Orthogonality - A First Course in Linear Algebra

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

##### math.la.t.vec.orthogonal_set.linindep
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# Orthogonality - A First Course in Linear Algebra

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

##### math.la.d.vec.orthonormal_set
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# Spanning Sets - A First Course in Linear Algebra

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

##### math.la.t.mat.null_space.rref.span
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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.scalar.mult
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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.

##### math.la.t.mat.eqn.lincomb
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# Matrix Operations - A First Course in Linear Algebra

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

##### math.la.d.mat.z
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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
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# Linear Independence - A First Course in Linear Algebra

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

##### math.la.t.vec.linindep.pivot
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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.

##### math.la.d.vec.lincomb.coord
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# Matrix Operations - A First Course in Linear Algebra

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

##### math.la.t.mat.sum.transpose
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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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# Linear Independence - A First Course in Linear Algebra

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

##### math.la.t.vec.linindep.homog
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# Orthogonality - A First Course in Linear Algebra

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

##### math.la.t.innerproduct.distributive.rncn
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# Orthogonality - A First Course in Linear Algebra

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

##### math.la.t.innerproduct.commutative.cn
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5 years ago
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# Orthogonality - A First Course in Linear Algebra

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

##### math.la.t.vec.innerproduct.norm
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5 years ago
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# Linear Independence - A First Course in Linear Algebra

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

##### math.la.t.equiv.col.linindep.rep
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5 years ago
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# Reduced Row-Echelon Form - A First Course in Linear Algebra

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

##### math.la.d.mat.pivot_col
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5 years ago
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# Orthogonality - A First Course in Linear Algebra

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

##### math.la.t.gramschmidt
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# Matrix Operations - A First Course in Linear Algebra

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

##### math.la.t.mat.transpose.involution
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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_set
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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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# Matrix Operations - A First Course in Linear Algebra

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

##### math.la.t.mat.scalar.conjugate
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# Matrix Operations - A First Course in Linear Algebra

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

##### math.la.t.mat.conjugate.involution
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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.unit.coord
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# Vector Operations - A First Course in Linear Algebra

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

##### math.la.d.vec.equal.coord
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# Matrix Operations - A First Course in Linear Algebra

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

##### math.la.t.mat.symmetric.square
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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.sum
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# Matrix Operations - A First Course in Linear Algebra

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

##### math.la.d.mat.thediagonal
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# Matrix Operations - A First Course in Linear Algebra

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

##### math.la.d.mat.symmetric
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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.equal
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# Matrix Operations - A First Course in Linear Algebra

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

##### math.la.d.mat.conjugate
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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.

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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.

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# Matrix Operations - A First Course in Linear Algebra

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

##### math.la.t.mat.scalar.transpose
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# Matrix Operations - A First Course in Linear Algebra

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

##### math.la.d.mat.skewsymmetric
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# Matrix Multiplication - A First Course in Linear Algebra

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

##### math.la.t.mat.vec.prod.unique
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# Matrix Multiplication - A First Course in Linear Algebra

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

##### math.la.d.mat.vec.prod
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# Matrix Inverses and Systems of Linear Equations - A First Course in Linear Algebra

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

##### math.la.d.mat.invertible
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# Matrix Operations - A First Course in Linear Algebra

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

##### math.la.t.matsum.conjugate
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# Matrix Multiplication - A First Course in Linear Algebra

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

##### math.la.d.mat.mult.col
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# Matrix Multiplication - A First Course in Linear Algebra

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

##### math.la.t.mat.mult.conjugate
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# Matrix Inverses and Nonsingular Matrices - A First Course in Linear Algebra

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

##### math.la.t.mat.prod.nonsingular
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# Matrix Multiplication - A First Course in Linear Algebra

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

##### math.la.t.mat.mult.assoc
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# Matrix Inverses and Systems of Linear Equations - A First Course in Linear Algebra

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

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

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

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

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

##### math.la.t.mat.unitary.innerproduct
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# Matrix Multiplication - A First Course in Linear Algebra

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

##### math.la.t.mat.mult.coord
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# Matrix Multiplication - A First Course in Linear Algebra

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

##### math.la.t.mat.mult.transpose
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5 years ago
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# Matrix Inverses and Nonsingular Matrices - A First Course in Linear Algebra

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

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

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

##### math.la.t.mat.unitary.inv
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# Matrix Operations - A First Course in Linear Algebra

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

##### math.la.t.mat.transpose.conjugate
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# Column and Row Spaces - A First Course in Linear Algebra

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

##### math.la.t.mat.col_space.pivot
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# Four Subsets - A First Course in Linear Algebra

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

##### math.la.c.mat.col_space.row_reduce
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# Matrix Multiplication - A First Course in Linear Algebra

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

##### math.la.t.mat.mult.distributive
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# Vector Spaces - A First Course in Linear Algebra

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

##### math.la.t.vsp.vector.negative
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# Orthogonality - A First Course in Linear Algebra

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

##### math.la.d.innerproduct.cn
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# Matrix Inverses and Systems of Linear Equations - A First Course in Linear Algebra

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

##### math.la.t.mat.inv.involution
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# Matrix Inverses and Systems of Linear Equations - A First Course in Linear Algebra

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

##### math.la.t.mat.inv.2x2
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# Four Subsets - A First Course in Linear Algebra

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

##### math.la.t.mat.erref.spaces
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# Vector Spaces - A First Course in Linear Algebra

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

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

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

##### math.la.t.vsp.scalar.mult.z
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# Vector Spaces - A First Course in Linear Algebra

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

##### math.la.t.vsp.z.unique
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# Subspaces - A First Course in Linear Algebra

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

##### math.la.d.vsp.subspace.arb
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# Column and Row Spaces - A First Course in Linear Algebra

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

##### math.la.t.mat.eqn.lincomb
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# Column and Row Spaces - A First Course in Linear Algebra

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

##### math.la.t.mat.row_space.pivot
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# Vector Spaces - A First Course in Linear Algebra

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

##### math.la.d.vec.arb
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# Vector Spaces - A First Course in Linear Algebra

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

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

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

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

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

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

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

##### math.la.t.mat.null_space.rref.basis
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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.

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# Similarity and Diagonalization - A First Course in Linear Algebra

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

##### math.la.t.mat.diagonalizable
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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.

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# Dimension - A First Course in Linear Algebra

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

##### math.la.d.vsp.dim.finite_infinite.arb
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# Linear Independence and Spanning Sets - A First Course in Linear Algebra

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

##### math.la.d.vec.linindep.arb
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# Matrix Multiplication - A First Course in Linear Algebra

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

##### math.la.t.mat.eqn.linsys
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# Dimension - A First Course in Linear Algebra

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

##### math.la.d.mat.nullity
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# Subspaces - A First Course in Linear Algebra

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

##### math.la.e.vsp.row
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# Linear Independence and Spanning Sets - A First Course in Linear Algebra

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

##### math.la.d.vec.lindep.relation.trvial.rep
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# Linear 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.mat.basis.standard.coord
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# Determinant of a Matrix - A First Course in Linear Algebra

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

##### math.la.t.mat.det.cofactor.row
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# Eigenvalues and Eigenvectors - A First Course in Linear Algebra

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

##### math.la.t.mat.eigsp.subsp
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# Dimension - A First Course in Linear Algebra

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

##### math.la.t.mat.rank.pivot
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# Properties of Dimension - A First Course in Linear Algebra

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

##### math.la.t.vsp.linindep.extend
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# Bases - A First Course in Linear Algebra

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

##### math.la.t.subspace.basis.orthogonal
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# Determinant of a Matrix - A First Course in Linear Algebra

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

##### math.la.t.mat.det.cofactor.col
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# Matrix Multiplication - A First Course in Linear Algebra

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

##### math.la.c.mat.mult
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# Properties of Dimension - A First Course in Linear Algebra

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

##### math.la.t.mat.erref.dimension
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# Subspaces - A First Course in Linear Algebra

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

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

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

##### math.la.t.equiv.col.basis
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# Properties of Dimension - A First Course in Linear Algebra

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

##### math.la.t.vsp.subspace.dim.arb
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# Determinant of a Matrix - A First Course in Linear Algebra

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

##### math.la.d.mat.det.cofactor
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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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# 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
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# Bases - A First Course in Linear Algebra

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

##### math.la.d.vsp.basis.standard.leq_n
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# Injective Linear Transformations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.injective.dim
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# Properties of Dimension - A First Course in Linear Algebra

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

##### math.la.t.vsp.subspace.dim.equal
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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.eigsp
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# Determinant of a Matrix - A First Course in Linear Algebra

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

##### math.la.d.mat.cofactor
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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.charpoly
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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.

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# Determinant of a Matrix - A First Course in Linear Algebra

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

##### math.la.d.mat.elementary.prod
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# Properties of Determinants of Matrices - A First Course in Linear Algebra

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

##### math.la.t.mat.det.scalar
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# Bases - A First Course in Linear Algebra

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

##### math.la.d.vsp.basis.arb
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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.equiv.det
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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
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# Similarity and Diagonalization - A First Course in Linear Algebra

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

##### math.la.t.mat.similar.equiv
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# Orthonormal Diagonalization - A First Course in Linear Algebra

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

##### math.la.d.mat.normal
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# Properties of Eigenvalues and Eigenvectors - A First Course in Linear Algebra

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

##### math.la.t.equiv.eig
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# Properties of Eigenvalues and Eigenvectors - A First Course in Linear Algebra

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

##### math.la.t.mat.eig.multiplicity.eigenspace
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# Similarity and Diagonalization - A First Course in Linear Algebra

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

##### math.la.d.mat.similar
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5 years ago
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# Properties of Eigenvalues and Eigenvectors - A First Course in Linear Algebra

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

##### math.la.t.mat.eig.inv
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# Matrix Multiplication - A First Course in Linear Algebra

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

##### math.la.t.mat.mult.z
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# Injective Linear Transformations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.equiv.basis
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# Matrix Multiplication - A First Course in Linear Algebra

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

##### math.la.t.innerproduct.mat.cn
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# Properties of Eigenvalues and Eigenvectors - A First Course in Linear Algebra

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

##### math.la.t.mat.eig.scalar
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# Linear Independence and Spanning Sets - A First Course in Linear Algebra

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

##### math.la.d.vsp.span.set.arb
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# Vector Spaces - A First Course in Linear Algebra

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

##### math.la.e.vsp.sequence
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# Surjective Linear Transformations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.basis.span.surjective.arb
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# Orthonormal Diagonalization - A First Course in Linear Algebra

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

##### math.la.t.mat.normal.diagonalize
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# Determinant of a Matrix - A First Course in Linear Algebra

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

##### math.la.c.mat.det
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# Four Subsets - A First Course in Linear Algebra

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

##### math.la.t.mat.erref.of
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# Matrix Multiplication - A First Course in Linear Algebra

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

##### math.la.d.mat.hermitian
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# Surjective Linear Transformations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.surjective.dim
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# Dimension - A First Course in Linear Algebra

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

##### math.la.t.equiv.nullity
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# Column and Row Spaces - A First Course in Linear Algebra

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

##### math.la.d.mat.row_space.row_equiv
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# Eigenvalues and Eigenvectors - A First Course in Linear Algebra

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

##### math.la.t.mat.eigsp.nullspace
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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
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# Vector Spaces - A First Course in Linear Algebra

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

##### math.la.d.vsp.vector.negative.unique
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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.elementary.det
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# Orthonormal Diagonalization - A First Course in Linear Algebra

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

##### math.la.t.mat.triangular.prod
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# Determinant of a Matrix - A First Course in Linear Algebra

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

##### math.la.t.mat.det.2x2
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# Matrix Inverses and Systems of Linear Equations - A First Course in Linear Algebra

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

##### math.la.t.mat.inv.shoesandsocks
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# Linear Transformations - A First Course in Linear Algebra

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

##### math.la.d.lintrans.scalar.arb
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# Matrix Multiplication - A First Course in Linear Algebra

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

##### math.la.t.mat.mult.identity
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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
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# Linear Transformations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.sum.arb
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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.d.vsp.basis.coord.vector.arb
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# Matrix Inverses and Systems of Linear Equations - A First Course in Linear Algebra

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

##### math.la.t.mat.inv.augmented
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# Eigenvalues and Eigenvectors - A First Course in Linear Algebra

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

##### math.la.t.mat.charpoly.eig
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# Vector Spaces - A First Course in Linear Algebra

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

##### math.la.e.vsp.crazy
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# Linear Transformations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.z
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# Matrix Inverses and Nonsingular Matrices - A First Course in Linear Algebra

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

##### math.la.d.mat.unitary
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# Determinant of a Matrix - A First Course in Linear Algebra

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

##### math.la.t.mat.mult.elementary
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# Linear Transformations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.lincomb
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# Surjective Linear Transformations - A First Course in Linear Algebra

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

##### math.la.d.lintrans.range.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.mat.null_space.rncn
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# Invertible Linear Transformations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.ranknullity
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# Invertible Linear Transformations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.surjective.rank
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# Change of Basis - A First Course in Linear Algebra

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

##### math.la.t.lintrans.mat_repn.eig
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# Properties of Determinants of Matrices - A First Course in Linear Algebra

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

##### math.la.t.mat.det.switch
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5 years ago
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# Vector Representations - A First Course in Linear Algebra

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

##### math.la.t.vsp.basis.coord.lin.arb
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# Invertible Linear Transformations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.composition.invertible.arb
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# Properties of Eigenvalues and Eigenvectors - A First Course in Linear Algebra

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

##### math.la.t.mat.hermitian.eigvec.orthogonal
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# Matrix Representations - A First Course in Linear Algebra

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

##### math.la.t.equiv.lintrans.inv
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# Invertible Linear Transformations - A First Course in Linear Algebra

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

##### math.la.d.vsp.isomorphism
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# Matrix Representations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.mat_repn.composition
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5 years ago
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# Linear Transformations - A First Course in Linear Algebra

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

##### math.la.c.transformation.matrix
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# Linear Transformations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.scalar.arb
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5 years ago
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# Properties of Eigenvalues and Eigenvectors - A First Course in Linear Algebra

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

##### math.la.t.mat.hermitian.eig.real
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5 years ago
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# Change of Basis - A First Course in Linear Algebra

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

##### math.la.d.lintrans.eigvec
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5 years ago
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# Change of Basis - A First Course in Linear Algebra

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

##### math.la.t.vsp.change_of_basis
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# Invertible Linear Transformations - A First Course in Linear Algebra

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

##### math.la.d.lintrans.rank
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5 years ago
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2
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# Linear Transformations - A First Course in Linear Algebra

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

##### math.la.d.lintrans.sum.arb
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5 years ago
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# Matrix Inverses and Systems of Linear Equations - A First Course in Linear Algebra

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

##### math.la.t.mat.inv.scalar
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September 11th, 2017
5 years ago
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2
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# Properties of Eigenvalues and Eigenvectors - A First Course in Linear Algebra

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

##### math.la.t.mat.eig.polynomial
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5 years ago
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# Orthonormal Diagonalization - A First Course in Linear Algebra

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

##### math.la.t.mat.normal.eigenval
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5 years ago
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# Invertible Linear Transformations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.inv.shoesandsocks
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# Injective Linear Transformations - A First Course in Linear Algebra

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

##### math.la.d.lintrans.kernel.arb
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5 years ago
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# Properties of Eigenvalues and Eigenvectors - A First Course in Linear Algebra

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

##### math.la.t.mat.eigvec.linindep
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5 years ago
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# Similarity and Diagonalization - A First Course in Linear Algebra

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

##### math.la.t.mat.diagonalizable.distinct
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# Surjective Linear Transformations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.range.span.arb
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# Linear Transformations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.composition.arb
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5 years ago
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2
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# Vector Representations - A First Course in Linear Algebra

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

##### math.la.t.vsp.dim.isomorphic
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5 years ago
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3
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# Linear Transformations - A First Course in Linear Algebra

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

##### math.la.d.lintrans.preimage.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.foundation.d.axiom
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# Bases - A First Course in Linear Algebra

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

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

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

##### math.la.d.lintrans.nullity
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5 years ago
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3
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# Similarity and Diagonalization - A First Course in Linear Algebra

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

##### math.la.t.mat.diagonalizable.charpoly
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5 years ago
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# Similarity and Diagonalization - A First Course in Linear Algebra

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

##### math.la.d.mat.diagonal
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5 years ago
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# Invertible Linear Transformations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.invertible.arb
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5 years ago
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# Injective Linear Transformations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.injective.linindep
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5 years ago
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# Injective Linear Transformations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.equiv.kernel
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# Matrix Multiplication - A First Course in Linear Algebra

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

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# Bases - A First Course in Linear Algebra

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

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

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

##### math.la.d.mat.rank
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# Vector Representations - A First Course in Linear Algebra

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

##### math.la.t.vsp.linindep.coord
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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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# Matrix Representations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.mat.repn.arb
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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.vsp.change_of_basis.arb
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# Similarity and Diagonalization - A First Course in Linear Algebra

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

##### math.la.d.mat.diagonalization
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# Change of Basis - A First Course in Linear Algebra

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

##### math.la.t.vsp.change_of_basis.inv
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# Matrix Representations - A First Course in Linear Algebra

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

##### math.la.d.lintrans.mat.repn.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.

##### math.la.t.vsp.basis.coord.surjective.arb
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# Matrix Multiplication - A First Course in Linear Algebra

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

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# Orthonormal Diagonalization - A First Course in Linear Algebra

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

##### math.la.t.lintrans.mat_repn.triangular
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# Matrix Multiplication - A First Course in Linear Algebra

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

##### math.la.d.mat.hermitian.innerproduct.cn
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# Matrix Inverses and Nonsingular Matrices - A First Course in Linear Algebra

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

##### math.la.t.equiv.inv
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# Change of Basis - A First Course in Linear Algebra

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

##### math.la.t.vsp.change_of_basis.conjugate
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# Properties of Eigenvalues and Eigenvectors - A First Course in Linear Algebra

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

##### math.la.d.mat.eig.number
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# Matrix Representations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.mat_repn.sum
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# Invertible Linear Transformations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.inv.involution.arb
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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
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# Vector Representations - A First Course in Linear Algebra

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

##### math.la.t.vsp.basis.coord.injective.arb
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# Matrix Inverses and Nonsingular Matrices - A First Course in Linear Algebra

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

##### math.la.t.mat.inv.oneside
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# Matrix Multiplication - A First Course in Linear Algebra

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

##### math.la.t.mat.scalar.prod.commut
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# Properties of Eigenvalues and Eigenvectors - A First Course in Linear Algebra

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

##### math.la.t.mat.real.eig.cn
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# Injective Linear Transformations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.preimage.translation.arb
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# Similarity and Diagonalization - A First Course in Linear Algebra

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

##### math.la.t.mat.similar.eig
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# Bases - A First Course in Linear Algebra

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

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

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

##### math.la.d.mat.null_space.left
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# Column and Row Spaces - A First Course in Linear Algebra

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

##### math.la.d.mat.col_space
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# Determinant of a Matrix - A First Course in Linear Algebra

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

##### math.la.d.mat.elementary.inv
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# Four Subsets - A First Course in Linear Algebra

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

##### math.la.d.mat.erref.of
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# Matrix Representations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.mat_repn.scalar
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# Injective Linear Transformations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.kernel.arb
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# Orthonormal Diagonalization - A First Course in Linear Algebra

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

##### math.la.t.mat.triangular.inv
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# Surjective Linear Transformations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.range.subsp.arb
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# Injective Linear Transformations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.composition.injective.arb
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# Invertible Linear Transformations - A First Course in Linear Algebra

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

##### math.la.d.lintrans.inv.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.

##### math.la.d.lintrans.composition.arb
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# Properties of Determinants of Matrices - A First Course in Linear Algebra

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

##### math.la.t.mat.det.product
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# Invertible Linear Transformations - A First Course in Linear Algebra

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

##### math.la.d.lintrans.identity
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# Linear Independence and Spanning Sets - A First Course in Linear Algebra

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

##### math.la.t.vsp.basis.coord.unique
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# Vector Spaces - A First Course in Linear Algebra

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

##### math.la.t.vsp.vector.mult.z
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# Dimension - A First Course in Linear Algebra

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

##### math.la.t.mat.ranknullity
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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
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# Dimension - A First Course in Linear Algebra

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

##### math.la.t.vsp.dim.more.lindep.arb
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# Properties of Eigenvalues and Eigenvectors - A First Course in Linear Algebra

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

##### math.la.t.mat.eig.power
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# Properties of Dimension - A First Course in Linear Algebra

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

##### math.la.t.mat.row_space.col_space
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# Properties of Eigenvalues and Eigenvectors - A First Course in Linear Algebra

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

##### math.la.t.mat.eig.transpose
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# Subspaces - A First Course in Linear Algebra

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

##### math.la.d.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.mat.null_space.left.rncn
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# Vector Spaces - A First Course in Linear Algebra

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

##### math.la.t.vsp.mult.z
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# Determinant of a Matrix - A First Course in Linear Algebra

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

##### math.la.d.mat.elementary
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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.

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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.

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# Surjective Linear Transformations - A First Course in Linear Algebra

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

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# Orthonormal Diagonalization - A First Course in Linear Algebra

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

##### math.la.t.mat.triangular.unitary
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# Orthonormal Diagonalization - A First Course in Linear Algebra

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

##### math.la.d.mat.triangular.upper
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# Injective Linear Transformations - A First Course in Linear Algebra

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

##### math.la.d.lintrans.injective.arb
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# Orthonormal Diagonalization - A First Course in Linear Algebra

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

##### math.la.d.mat.triangular.upper
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# Surjective Linear Transformations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.composition.surjective.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.

##### math.la.d.lintrans.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.

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# Subspaces - A First Course in Linear Algebra

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

##### math.la.d.vec.lincomb.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.z
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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.

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

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

##### math.la.t.equiv.col.span
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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.

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# Linear Transformations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.vsp
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# Linear Transformations - A First Course in Linear Algebra

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

##### math.la.t.lintrans.basis
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# Matrix Representations - A First Course in Linear Algebra

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

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# The cross product - Math Insight

A rotatable model of the cross product of two vectors.