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Understanding Linear Algebra
David Austin
Contents
Index
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Contents
Prev
Up
Next
Front Matter
Dedication
Colophon
Our goals
1
Systems of equations
What can we expect
Finding solutions to linear systems
Computation with Sage
Pivots and their influence on solution spaces
2
Vectors, matrices, and linear combinations
Vectors and linear combinations
Matrix multiplication and linear combinations
The span of a set of vectors
Linear independence
Matrix transformations
The geometry of matrix transformations
3
Invertibility, bases, and coordinate systems
Invertibility
Bases and coordinate systems
Image compression
Determinants
Subspaces
4
Eigenvalues and eigenvectors
An introduction to eigenvalues and eigenvectors
Finding eigenvalues and eigenvectors
Diagonalization, similarity, and powers of a matrix
Dynamical systems
Markov chains and Google's PageRank algorithm
5
Linear algebra and computing
Gaussian elimination revisited
Finding eigenvectors numerically
6
Orthogonality and Least Squares
The dot product
Orthogonal complements and the matrix transpose
Orthogonal bases and projections
Finding orthogonal bases
Orthogonal least squares
7
The Spectral Theorem and singular value decompositions
Symmetric matrices and variance
Quadratic forms
Principal Component Analysis
Singular Value Decompositions
Using Singular Value Decompositions
Back Matter
A
Sage Reference
Index
Colophon
Authored in PreTeXt
Colophon
Colophon
This book was authored in PreTeXt.