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Linear Algebra with SageMath:
A Gentle Introduction
Ajit Kumar
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\( \newcommand{\Loadedframemethod}{default} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\inner}[2]{\left\langle #1,#2\right\rangle} \newcommand{\norm}[1]{\left\lVert#1\right\rVert} \newcommand{\normx}[1]{\left\Vert#1\right\Vert} \newcommand{\partd}[2]{\dfrac{\partial #1}{\partial #2}} \newcommand{\innprod}[2]{\left\lt #1,#2\right>} \def\rank{{ rank\,}} \newcommand{\ds}{\displaystyle} \def\diag{{ diag\,}} \def\proj{{ proj\,}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)
Front Matter
Colophon
Author Biography
Dedication
1
System of Linear Equations
1.1
Elementary Row Operations
1.1.1
Elmentary Row Operations
1.1.2
Matrix Inversion via Elementary Row Operations
1.2
Elementary Column Operations
1.3
Echelon Forms
1.3.1
Row Echelon Form
1.3.2
Gaussian Elimination Method
1.3.3
Gauss-Jordan elimination method
1.4
Rank of Matrices
1.5
Homogeneous System of Linear Equations
1.6
\(LU\)
-Facotorization
1.6.1
Dooliitle and Crout’s Factorization
1.6.2
Solving system of equations using LU factorization
1.6.3
\(LU\)
-factorization in Sage
1.6.4
User defined functions for DooLitlte and Crout’s Methods
1.7
Exercises
2
\(\R^n\)
as a Vector Space
2.1
Introduction
2.2
Linear Spans
2.3
Linear Dependence
2.4
Basis and Dimension
2.4.1
Basis
2.4.2
Dimension of Subspaces
2.4.3
Change of bases.
2.5
Sage Computations
3
Linear Transformations
3.1
Introduction
3.2
Linear maps from
\(\R^n\)
to
\(\R^m\)
3.2.1
Linear maps from
\(\R^n\)
to
\(\R^m\)
3.2.2
Composition of linear transformations
3.2.3
Matrix of Change of basis
3.2.3
Reading Questions
3.3
Reflections and Projections
3.3.1
Reflections in
\(\R^2\)
3.3.2
Projections in
\(\R^2\)
3.3.3
Projection and Reflection in
\(\R^3\)
3.4
Geometry of Linear Transformations
3.5
Sage Computations
4
Vector Spaces
4.1
Introduction
4.2
Vector Subspaces
4.3
Linear Span
4.4
Linear dependence and independence
4.5
Basis and dimension
4.5.1
Basis of a Vector Space
4.5.2
How to find a basis of a finite dimensional vector space?
4.5.3
Lagrange Interpolation
4.5.4
Dimension Formula
4.6
Sage Computations
4.7
Exercise Set
5
Eigenvalues and Eigenvectors
5.1
Eigenvalues and Eigenvectors
5.1
Reading Questions
5.2
Diagonalization
5.3
Applications of Eigenvalues and Eigenvectors
5.3.1
Fibonacci Sequence
5.3.2
Predator-Pray Model
5.3.3
Solving System of Linear ODE
5.3.4
Markov Chains
5.3.5
Google Search Engine
5.4
Exercises on Eigenvalues and Eigenvectors
6
Orthogonality
6.1
Orthogonality
6.2
Gram-Schmidt Orthogonalization Process
6.3
Orthogonal Complements
6.4
Orthogonal Diagonalizations
6.5
QR-Factorization
7
Inner Product
7.1
Inner Product
7.2
Exercise Set
8
Least Square Problems
8.1
Least Square Problems
8.1.1
Linear Least Square Problems
8.1.1
Exercises
8.1.2
Fitting polynomials to a data set
8.1.2
Exercises
8.1.3
Weighted Least Square Problems
9
Singular Value Decomposition
9.1
Singular Value Decomposition
9.1.1
Singular Value Decomposition Theorem
9.1.1
Reading Questions
9.1.1
Exercises
9.1.1
Exercises
9.1.2
Pseudoinverse using SVD
9.1.2
Reading Questions
9.1.2
Exercises
9.1.3
Geometry of SVD
9.1.4
Image Compression using SVD
10
Principal Component Analysis
10.1
Principal Component Analysis
10.1.1
Introduction
10.1.2
Mathematics behind PCA
10.2
Applications of PCA
10.2.1
Image compression with PCA
10.2.2
Relation Between SVD and PCA
10.2.3
Exercise Set
10.3
Sage Practice Area
10.3.1
Sage Practice Area
Backmatter
Colophon
Colophon
This book was authored in PreTeXt.
