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Understanding Linear Algebra
David Austin
Contents
Index
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Contents
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Front Matter
Dedication
Colophon
Our goals
1
Systems of equations
What can we expect
Finding solutions to systems of linear equations
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 of \(\real^p\)
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
The dot product
The tranpose and orthogonality
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
The Singular Value Decomposition
Using Singular Value Decompositions
Back Matter
Index
Authored in PreTeXt
Understanding Linear Algebra
David Austin
Department of Mathematics
Grand Valley State University
austind@gvsu.edu
August 16, 2020
Dedication
Colophon
Our goals
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