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Understanding Linear Algebra
David Austin
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Front Matter
Colophon
Dedication
Acknowledgements
Our goals
A note on the print version
1
Systems of equations
1.1
What can we expect
1.1.1
Some simple examples
1.1.2
Systems of linear equations
1.1.3
Summary
1.2
Finding solutions to linear systems
1.2.1
Gaussian elimination
1.2.2
Augmented matrices
1.2.3
Reduced row echelon form
1.2.4
Summary
1.2.5
Exercises
1.3
Computation with Sage
1.3.1
Introduction to Sage
1.3.2
Sage and matrices
1.3.3
Computational effort
1.3.4
Summary
1.3.5
Exercises
1.4
Pivots and their influence on solution spaces
1.4.1
The existence of solutions
1.4.2
The uniqueness of solutions
1.4.3
Summary
1.4.4
Exercises
2
Vectors, matrices, and linear combinations
2.1
Vectors and linear combinations
2.1.1
Vectors
2.1.2
Linear combinations
2.1.3
Summary
2.1.4
Exercises
2.2
Matrix multiplication and linear combinations
2.2.1
Scalar multiplication and addition of matrices
2.2.2
Matrix-vector multiplication and linear combinations
2.2.3
Matrix-vector multiplication and linear systems
2.2.4
Matrix-matrix products
2.2.5
Summary
2.2.6
Exercises
2.3
The span of a set of vectors
2.3.1
The span of a set of vectors
2.3.2
Pivot positions and span
2.3.3
Summary
2.3.4
Exercises
2.4
Linear independence
2.4.1
Linear dependence
2.4.2
How to recognize linear dependence
2.4.3
Homogeneous equations
2.4.4
Summary
2.4.5
Exercises
2.5
Matrix transformations
2.5.1
Matrix transformations
2.5.2
Composing matrix transformations
2.5.3
Discrete Dynamical Systems
2.5.4
Summary
2.5.5
Exercises
2.6
The geometry of matrix transformations
2.6.1
The geometry of
2
×
2
matrix transformations
2.6.2
Matrix transformations and computer animation
2.6.3
Summary
2.6.4
Exercises
3
Invertibility, bases, and coordinate systems
3.1
Invertibility
3.1.1
Invertible matrices
3.1.2
Solving equations with an inverse
3.1.3
Triangular matrices and Gaussian elimination
3.1.4
Summary
3.1.5
Exercises
3.2
Bases and coordinate systems
3.2.1
Bases
3.2.2
Coordinate systems
3.2.3
Examples of bases
3.2.4
Summary
3.2.5
Exercises
3.3
Image compression
3.3.1
Color models
3.3.2
The JPEG compression algorithm
3.3.3
Summary
3.3.4
Exercises
3.4
Determinants
3.4.1
Determinants of
2
×
2
matrices
3.4.2
Determinants and invertibility
3.4.3
Cofactor expansions
3.4.4
Summary
3.4.5
Exercises
3.5
Subspaces
3.5.1
Subspaces
3.5.2
The column space of
A
3.5.3
The null space of
A
3.5.4
Summary
3.5.5
Exercises
4
Eigenvalues and eigenvectors
4.1
An introduction to eigenvalues and eigenvectors
4.1.1
A few examples
4.1.2
The usefulness of eigenvalues and eigenvectors
4.1.3
Summary
4.1.4
Exercises
4.2
Finding eigenvalues and eigenvectors
4.2.1
The characteristic polynomial
4.2.2
Finding eigenvectors
4.2.3
The characteristic polynomial and the dimension of eigenspaces
4.2.4
Using Sage to find eigenvalues and eigenvectors
4.2.5
Summary
4.2.6
Exercises
4.3
Diagonalization, similarity, and powers of a matrix
4.3.1
Diagonalization of matrices
4.3.2
Powers of a diagonalizable matrix
4.3.3
Similarity and complex eigenvalues
4.3.4
Summary
4.3.5
Exercises
4.4
Dynamical systems
4.4.1
A first example
4.4.2
Classifying dynamical systems
4.4.3
A
3
×
3
system
4.4.4
Summary
4.4.5
Exercises
4.5
Markov chains and Google’s PageRank algorithm
4.5.1
A first example
4.5.2
Markov chains
4.5.3
Google’s PageRank algorithm
4.5.4
Summary
4.5.5
Exercises
5
Linear algebra and computing
5.1
Gaussian elimination revisited
5.1.1
Partial pivoting
5.1.2
L
U
factorizations
5.1.3
Summary
5.1.4
Exercises
5.2
Finding eigenvectors numerically
5.2.1
The power method
5.2.2
Finding other eigenvalues
5.2.3
Summary
5.2.4
Exercises
6
Orthogonality and Least Squares
6.1
The dot product
6.1.1
The geometry of the dot product
6.1.2
k
-means clustering
6.1.3
Summary
6.1.4
Exercises
6.2
Orthogonal complements and the matrix transpose
6.2.1
Orthogonal complements
6.2.2
The matrix transpose
6.2.3
Properties of the matrix transpose
6.2.4
Summary
6.2.5
Exercises
6.3
Orthogonal bases and projections
6.3.1
Orthogonal sets
6.3.2
Orthogonal projections
6.3.3
Summary
6.3.4
Exercises
6.4
Finding orthogonal bases
6.4.1
Gram-Schmidt orthogonalization
6.4.2
Q
R
factorizations
6.4.3
Summary
6.4.4
Exercises
6.5
Orthogonal least squares
6.5.1
A first example
6.5.2
Solving least-squares problems
6.5.3
Using
Q
R
factorizations
6.5.4
Polynomial Regression
6.5.5
Summary
6.5.6
Exercises
7
Singular value decompositions
7.1
Symmetric matrices and variance
7.1.1
Symmetric matrices and orthogonal diagonalization
7.1.2
Variance
7.1.3
Summary
7.1.4
Exercises
7.2
Quadratic forms
7.2.1
Quadratic forms
7.2.2
Definite symmetric matrices
7.2.3
Summary
7.2.4
Exercises
7.3
Principal Component Analysis
7.3.1
Principal Component Analysis
7.3.2
Using Principal Component Analysis
7.3.3
Summary
7.3.4
Exercises
7.4
Singular Value Decompositions
7.4.1
Finding singular value decompositions
7.4.2
The structure of singular value decompositions
7.4.3
Reduced singular value decompositions
7.4.4
Summary
7.4.5
Exercises
7.5
Using Singular Value Decompositions
7.5.1
Least-squares problems
7.5.2
Rank
k
approximations
7.5.3
Principal component analysis
7.5.4
Image compressing and denoising
7.5.5
Analyzing Supreme Court cases
7.5.6
Summary
7.5.7
Exercises
Back Matter
A
Sage Reference
Index
Colophon
Understanding Linear Algebra
David Austin
Department of Mathematics
Grand Valley State University
austind@gvsu.edu
August 9, 2025
Colophon
Dedication
Acknowledgements
Our goals
A note on the print version
🔗
