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Linear Algebra:
A second course, featuring proofs and Python
Sean Fitzpatrick
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\(\newcommand{\spn}{\operatorname{span}} \newcommand{\bbm}{\begin{bmatrix}} \newcommand{\ebm}{\end{bmatrix}} \newcommand{\R}{\mathbb{R}} \ifdefined\C \renewcommand\C{\mathbb{C}} \else \newcommand\C{\mathbb{C}} \fi \newcommand{\im}{\operatorname{im}} \newcommand{\nll}{\operatorname{null}} \newcommand{\csp}{\operatorname{col}} \newcommand{\rank}{\operatorname{rank}} \newcommand{\diag}{\operatorname{diag}} \newcommand{\tr}{\operatorname{tr}} \newcommand{\dotp}{\!\boldsymbol{\cdot}\!} \newcommand{\len}[1]{\lVert #1\rVert} \newcommand{\abs}[1]{\lvert #1\rvert} \newcommand{\proj}[2]{\operatorname{proj}_{#1}{#2}} \newcommand{\bz}{\overline{z}} \newcommand{\zz}{\mathbf{z}} \newcommand{\uu}{\mathbf{u}} \newcommand{\vv}{\mathbf{v}} \newcommand{\ww}{\mathbf{w}} \newcommand{\xx}{\mathbf{x}} \newcommand{\yy}{\mathbf{y}} \newcommand{\zer}{\mathbf{0}} \newcommand{\vecq}{\mathbf{q}} \newcommand{\vecp}{\mathbf{p}} \newcommand{\vece}{\mathbf{e}} \newcommand{\basis}[2]{\{\mathbf{#1}_1,\mathbf{#1}_2,\ldots,\mathbf{#1}_{#2}\}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)
Front Matter
Colophon
Preface
1
Vector spaces
1.1
Definition and examples
1.1
Exercises
1.2
Properties
1.3
Subspaces
1.3
Exercises
1.4
Span
1.4
Exercises
1.5
Project: working with span
1.6
Linear Independence
1.6
Exercises
1.7
Basis and dimension
1.7
Exercises
1.8
New subspaces from old
1.8
Exercises
2
Linear Transformations
2.1
Definition and examples
2.1
Exercises
2.2
Kernel and Image
2.2
Exercises
2.3
Isomorphisms, composition, and inverses
2.3.1
Isomorphisms
2.3.2
Composition and inverses
2.3.2
Exercises
2.4
Project: matrix transformations
2.5
Project: linear recurrences
3
Orthogonality and Applications
3.1
Orthogonal sets of vectors
3.1.1
Basic definitions and properties
3.1.2
Orthogonal sets of vectors
3.1.3
Exercises
3.2
The Gram-Schmidt Procedure
3.2
Exercises
3.3
Orthogonal Projection
3.3
Exercises
3.4
Project: Orthogonal bases
3.5
Project: dual basis
3.6
Project: Least squares approximation
4
Diagonalization
4.1
Eigenvalues and Eigenvectors
4.1
Exercises
4.2
Diagonalization of symmetric matrices
4.2
Exercises
4.3
Quadratic forms
4.3
Exercises
4.4
Diagonalization of complex matrices
4.4.1
Complex vectors
4.4.2
Complex matrices
4.4.3
Exercises
4.5
Project: linear dynamical systems
4.6
Matrix Factorizations and Eigenvalues
4.6.1
Matrix Factorizations
4.6.1.1
Positive Operators
4.6.1.2
Singular Value Decomposition
4.6.1.3
QR Factorization
4.6.2
Computing Eigenvalues
4.6.2.1
The Power Method
4.6.2.2
The QR Algorithm
4.6.3
Exercises
4.7
Project: Eigenvalues and diagonalization
4.7.1
Orthogonal diagonalization
4.7.2
The power method for dominant eigenvalues
4.8
Project: Singular Value Decomposition
5
Change of Basis
5.1
The matrix of a linear transformation
5.1
Exercises
5.2
The matrix of a linear operator
5.2
Exercises
5.3
Project: bilinear forms
5.4
Direct Sums and Invariant Subspaces
5.4.1
Invariant subspaces
5.4.2
Eigenspaces
5.4.3
Direct Sums
5.5
Project: generalized eigenvectors
5.6
Generalized eigenspaces
5.7
Jordan Canonical Form
5.7
Exercises
Back Matter
A
Review of complex numbers
B
Computational Tools
B.1
Jupyter
B.2
Python basics
B.3
SymPy for linear algebra
B.3.1
SymPy basics
B.3.2
Matrices in SymPy
C
Solutions to Selected Exercises
Colophon
Colophon
Edition
Version 2.1.0 (now with Runestone)
©2024 Sean Fitzpatrick
Licensed to the public under Creative Commons Attribution-ShareAlike 4.0 International Public License
