From b091dc24b15d5c8a8a5d0f655f6c2e36a3fb11cb Mon Sep 17 00:00:00 2001 From: Alex Clarke Date: Mon, 8 Jan 2024 16:13:21 -0700 Subject: [PATCH] Initial commit of some initial documentation --- .gitignore | 303 ++++++++++++++++++ Linear-Algebra-Theorems-and-Definitions.pdf | Bin 0 -> 146340 bytes Linear-Algebra-Theorems-and-Definitions.tex | 34 ++ chapter-0.tex | 88 +++++ chapter-1/bases-and-dimension.tex | 1 + chapter-1/chapter-1.tex | 8 + chapter-1/introduction.tex | 1 + ...ations-and-systems-of-linear-equations.tex | 1 + ...ear-dependence-and-linear-independence.tex | 1 + .../maximal-linearly-independent-subsets.tex | 1 + chapter-1/subspaces.tex | 1 + chapter-1/vector-spaces.tex | 1 + chapter-2/chapter-2.tex | 8 + ...nsformations-and-matrix-multiplication.tex | 1 + chapter-2/dual-spaces.tex | 1 + ...l-equations-with-constant-coefficients.tex | 1 + chapter-2/invertibility-and-isomorphisms.tex | 1 + ...transformations-null-spaces-and-ranges.tex | 1 + chapter-2/the-change-of-coordinate-matrix.tex | 1 + ...resentation-of-a-linear-transformation.tex | 1 + chapter-3/chapter-3.tex | 5 + ...rix-operations-and-elementary-matrices.tex | 1 + ...linear-equations-computational-aspects.tex | 1 + ...f-linear-equations-theoretical-aspects.tex | 1 + ...e-rank-of-a-matrix-and-matrix-inverses.tex | 1 + .../a-characterization-of-the-determinant.tex | 1 + chapter-4/chapter-4.tex | 6 + chapter-4/determinants-of-order-2.tex | 1 + chapter-4/determinants-of-order-n.tex | 1 + chapter-4/properties-of-determinants.tex | 1 + ...ary-important-facts-about-determinants.tex | 1 + chapter-5/chapter-5.tex | 5 + chapter-5/diagonalizability.tex | 1 + chapter-5/eigenvalues-and-eigenvectors.tex | 1 + ...spaces-and-the-cayley-hamilton-theorem.tex | 1 + chapter-5/matrix-limits-and-markov-chains.tex | 1 + chapter-6/bilinear-and-quadratic-forms.tex | 1 + chapter-6/chapter-6.tex | 12 + ...conditioning-and-the-rayleigh-quotient.tex | 1 + ...einsteins-special-theory-of-relativity.tex | 1 + chapter-6/inner-products-and-norms.tex | 1 + .../normal-and-self-adjoint-operators.tex | 1 + ...l-projections-and-the-spectral-theorem.tex | 1 + .../the-adjoint-of-a-linear-operator.tex | 1 + .../the-geometry-of-orthogonal-operators.tex | 1 + ...ion-process-and-orthogonal-complements.tex | 1 + ...ue-decomposition-and-the-pseudoinverse.tex | 1 + ...rthogonal-operators-and-their-matrices.tex | 1 + chapter-7/chapter-7.tex | 5 + chapter-7/the-jordan-canonical-form-i.tex | 1 + chapter-7/the-jordan-canonical-form-ii.tex | 1 + chapter-7/the-minimal-polynomial.tex | 1 + chapter-7/the-rational-canonical-form.tex | 1 + init.sty | 59 ++++ 54 files changed, 575 insertions(+) create mode 100644 .gitignore create mode 100644 Linear-Algebra-Theorems-and-Definitions.pdf create mode 100644 Linear-Algebra-Theorems-and-Definitions.tex create mode 100644 chapter-0.tex create mode 100644 chapter-1/bases-and-dimension.tex create mode 100644 chapter-1/chapter-1.tex create mode 100644 chapter-1/introduction.tex create mode 100644 chapter-1/linear-combinations-and-systems-of-linear-equations.tex create mode 100644 chapter-1/linear-dependence-and-linear-independence.tex create mode 100644 chapter-1/maximal-linearly-independent-subsets.tex create mode 100644 chapter-1/subspaces.tex create mode 100644 chapter-1/vector-spaces.tex create mode 100644 chapter-2/chapter-2.tex create mode 100644 chapter-2/composition-of-linear-transformations-and-matrix-multiplication.tex create mode 100644 chapter-2/dual-spaces.tex create mode 100644 chapter-2/homogeneous-linear-differential-equations-with-constant-coefficients.tex create mode 100644 chapter-2/invertibility-and-isomorphisms.tex create mode 100644 chapter-2/linear-transformations-null-spaces-and-ranges.tex create mode 100644 chapter-2/the-change-of-coordinate-matrix.tex create mode 100644 chapter-2/the-matrix-representation-of-a-linear-transformation.tex create mode 100644 chapter-3/chapter-3.tex create mode 100644 chapter-3/elementary-matrix-operations-and-elementary-matrices.tex create mode 100644 chapter-3/systems-of-linear-equations-computational-aspects.tex create mode 100644 chapter-3/systems-of-linear-equations-theoretical-aspects.tex create mode 100644 chapter-3/the-rank-of-a-matrix-and-matrix-inverses.tex create mode 100644 chapter-4/a-characterization-of-the-determinant.tex create mode 100644 chapter-4/chapter-4.tex create mode 100644 chapter-4/determinants-of-order-2.tex create mode 100644 chapter-4/determinants-of-order-n.tex create mode 100644 chapter-4/properties-of-determinants.tex create mode 100644 chapter-4/summary-important-facts-about-determinants.tex create mode 100644 chapter-5/chapter-5.tex create mode 100644 chapter-5/diagonalizability.tex create mode 100644 chapter-5/eigenvalues-and-eigenvectors.tex create mode 100644 chapter-5/invariant-subspaces-and-the-cayley-hamilton-theorem.tex create mode 100644 chapter-5/matrix-limits-and-markov-chains.tex create mode 100644 chapter-6/bilinear-and-quadratic-forms.tex create mode 100644 chapter-6/chapter-6.tex create mode 100644 chapter-6/conditioning-and-the-rayleigh-quotient.tex create mode 100644 chapter-6/einsteins-special-theory-of-relativity.tex create mode 100644 chapter-6/inner-products-and-norms.tex create mode 100644 chapter-6/normal-and-self-adjoint-operators.tex create mode 100644 chapter-6/orthogonal-projections-and-the-spectral-theorem.tex create mode 100644 chapter-6/the-adjoint-of-a-linear-operator.tex create mode 100644 chapter-6/the-geometry-of-orthogonal-operators.tex create mode 100644 chapter-6/the-gram-schmidt-orthogonalization-process-and-orthogonal-complements.tex create mode 100644 chapter-6/the-singular-value-decomposition-and-the-pseudoinverse.tex create mode 100644 chapter-6/unitary-and-orthogonal-operators-and-their-matrices.tex create mode 100644 chapter-7/chapter-7.tex create mode 100644 chapter-7/the-jordan-canonical-form-i.tex create mode 100644 chapter-7/the-jordan-canonical-form-ii.tex create mode 100644 chapter-7/the-minimal-polynomial.tex create mode 100644 chapter-7/the-rational-canonical-form.tex create mode 100644 init.sty diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..7f7bed5 --- /dev/null +++ b/.gitignore @@ -0,0 +1,303 @@ +## Core latex/pdflatex auxiliary files: +*.aux +*.lof +*.log +*.lot +*.fls +*.out +*.toc +*.fmt +*.fot +*.cb +*.cb2 +.*.lb + +## Intermediate documents: +*.dvi +*.xdv +*-converted-to.* +# these rules might exclude image files for figures etc. +# *.ps +# *.eps +# *.pdf + +## Bibliography auxiliary files (bibtex/biblatex/biber): +*.bbl +*.bcf +*.blg +*-blx.aux +*-blx.bib +*.run.xml + +## Build tool auxiliary files: +*.fdb_latexmk +*.synctex +*.synctex(busy) +*.synctex.gz +*.synctex.gz(busy) +*.pdfsync + +## Build tool directories for auxiliary files +# latexrun +latex.out/ + +## Auxiliary and intermediate files from other packages: +# algorithms +*.alg +*.loa + +# achemso +acs-*.bib + +# amsthm +*.thm + +# beamer +*.nav +*.pre +*.snm +*.vrb + +# changes +*.soc + +# comment +*.cut + +# cprotect +*.cpt + +# elsarticle (documentclass of Elsevier journals) +*.spl + +# endnotes +*.ent + +# fixme +*.lox + +# feynmf/feynmp +*.mf +*.mp +*.t[1-9] +*.t[1-9][0-9] +*.tfm + +#(r)(e)ledmac/(r)(e)ledpar +*.end +*.?end +*.[1-9] +*.[1-9][0-9] +*.[1-9][0-9][0-9] +*.[1-9]R +*.[1-9][0-9]R +*.[1-9][0-9][0-9]R +*.eledsec[1-9] +*.eledsec[1-9]R +*.eledsec[1-9][0-9] +*.eledsec[1-9][0-9]R +*.eledsec[1-9][0-9][0-9] +*.eledsec[1-9][0-9][0-9]R + +# glossaries +*.acn +*.acr +*.glg +*.glo +*.gls +*.glsdefs +*.lzo +*.lzs +*.slg +*.slo +*.sls + +# uncomment this for glossaries-extra (will ignore makeindex's style files!) +# *.ist + +# gnuplot +*.gnuplot +*.table + +# gnuplottex +*-gnuplottex-* + +# gregoriotex +*.gaux +*.glog +*.gtex + +# htlatex +*.4ct +*.4tc +*.idv +*.lg +*.trc +*.xref + +# hyperref +*.brf + +# knitr 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+\documentclass[12pt,letterpaper]{report} +\usepackage{init} +\usepackage{import} +\usepackage{dsfont} + +\newcommand{\cond}[1]{\text{cond}\left(#1\right)} +\newcommand{\linear}[1]{\mathcal{L}\left(#1\right)} +\newcommand{\Id}{\mathds{I}} +\newcommand{\per}[1]{\text{per}\left(#1\right)} + +\setcounter{chapter}{-1} +\author{Alexander J. Clarke} +\title{Linear Algebra Theorems and Definitions} +\begin{document} +\maketitle +\clearpage +\begin{center} + \thispagestyle{empty} + \vspace*{\fill} + All theorems, corollaries, lemmas, remarks, and asides are direct quotes from Linear Algebra, 4th Edition, by Stephen H. Friedberg, Arnold J. Insel, and Lawrence E. Spence + \vspace*{\fill} +\end{center} +\tableofcontents + +\import{./}{chapter-0.tex} +\import{chapter-1/}{chapter-1.tex} +\import{chapter-2/}{chapter-2.tex} +\import{chapter-3/}{chapter-3.tex} +\import{chapter-4/}{chapter-4.tex} +\import{chapter-5/}{chapter-5.tex} +\import{chapter-6/}{chapter-6.tex} +\import{chapter-7/}{chapter-7.tex} + +\end{document} diff --git a/chapter-0.tex b/chapter-0.tex new file mode 100644 index 0000000..14ff366 --- /dev/null +++ b/chapter-0.tex @@ -0,0 +1,88 @@ +\chapter{List of Symbols} +\begin{align*} + & A_{ij} & \text{the $ij$-th entry of the matrix $A$} \\ + & A^{-1} & \text{the inverse of the matrix $A$} \\ + & A^\dag & \text{the pseudoinverse of the matrix $A$} \\ + & A^* & \text{the adjoint of the matrix $A$} \\ + & \tilde{A}_{ij} & \text{the matrix $A$ with row $i$ and column $j$ deleted} \\ + & A^t & \text{the transpose of the matrix $A$} \\ + & (A|B) & \text{the matrix $A$ augmented by the matrix $B$} \\ + & B_1 \bigoplus \dots \bigoplus B_k & \text{the direct sum of matrices $B_1$ through $B_k$} \\ + & \mathcal{B}(V) & \text{the set of bilinear forms on $V$} \\ + & \beta^* & \text{the dual basis of $\beta$} \\ + & \beta_x & \text{the $T$-cyclic basis generated by $x$} \\ + & \C & \text{the field of complex numbers} \\ + & \C_i & \text{the $i$th Gerschgorin disk} \\ + & \cond{A} & \text{the condition number of the matrix $A$} \\ + & C^n(\R) & \text{set of functions $f$ on $\R$ with $f^{(n)}$ continuous} \\ + & C^\infty & \text{set of functions with derivatives of every order} \\ + & C(\R) & \text{the vector space of continuous functions on $\R$} \\ + & C([0,1]) & \text{the vector space of continuous functions on $[0,1]$} \\ + & C_x & \text{the $T$-cyclic subspaces generated by $x$} \\ + & D & \text{the derivative operator on $C^\infty$} \\ + & \ldet{A} & \text{the determinant of the matrix $A$} \\ + & \delta_{ij} & \text{the Kronecker delta} \\ + & \ldim{V} & \text{the dimension of $V$} \\ + & e^A & \lim_{m \to \infty} \left(I + A + \frac{A^2}{2!} + \dots + \frac{A^m}{m!}\right) \\ + & e_i & \text{the $i$th standard vector of $\F^n$} \\ +\end{align*} +\begin{align*} + & E_\lambda & \text{the eigenspace of $T$ corresponding to $\lambda$} \\ + & \F & \text{a field} \\ + & f(A) & \text{the polynomial $f(x)$ evaluated at the matrix $A$} \\ + & F^n & \text{the set of $n$-tuples with entries in a field $\F$} \\ + & f(T) & \text{the polynomial $f(x)$ evaluated at the operator $T$} \\ + & \mathcal{F}(S,\F) & \text{the set of functions from $S$ to a field $\F$} \\ + & H & \text{space of continuous complex functions on $[0, 2\pi]$} \\ + & I_n \text{ or } I & \text{the $n \times n$ identity matrix} \\ + & \Id_V \text{ or } \Id & \text{the identity operator on $V$} \\ + & K_\lambda & \text{generalized eigenspace of $T$ corresponding to $\lambda$} \\ + & K_\phi & \{x\ |\ (\phi(T))^p(x) = 0 \text{, for some positive integer $p$}\} \\ + & L_A & \text{left-multiplication transformation by matrix $A$} \\ + & \lim_{m \to \infty}A_m & \text{the limit of a sequence of matrices} \\ + & \linear{V} & \text{the space of linear transformations from $V$ to $V$} \\ + & \linear{V, W} & \text{the space of linear transformations from $V$ to $W$} \\ + & M_{m \times n}(\F) & \text{the set of $m \times n$ matrices with entries in $\F$} \\ + & v(A) & \text{the column sum of the matrix $A$} \\ + & v_j(A) & \text{the $j$th column sum of the matrix $A$} \\ + & N(T) & \text{the null space of $T$} \\ + & \nullity{T} & \text{the dimension of the null space of $T$} \\ + & O & \text{the zero matrix} \\ + & \per{M} & \text{the permanent of the $2 \times 2$ matrix $M$} \\ + & P(\F) & \text{the space of polynomials with coefficients in $\F$} \\ + & P_n(\F) & \text{the polynomials in $P(\F)$ of degree at most $n$} \\ + & \phi_\beta & \text{the standard representation with respect to basis $\beta$} \\ + & \R & \text{the field of real numbers} \\ + & \rank{A} & \text{the rank of the matrix $A$} \\ + & \rank{T} & \text{the rank of the linear transformation $T$} \\ + & \rho(A) & \text{the row sum of the matrix $A$} \\ + & \rho_i(A) & \text{the $i$th row sum of the matrix $A$} \\ + & R(T) & \text{the range of the linear transformation $T$} \\ + & S_1 + S_2 & \text{the sum of sets $S_1$ and $S_2$} \\ + & \lspan{S} & \text{the span of the set $S$} \\ + & S^\perp & \text{the orthogonal complement of the set $S$} \\ + & [T]_\beta & \text{the matrix representation of $T$ in basis $\beta$} \\ + & [T]_\beta^\gamma & \text{the matrix representation of $T$ in bases $\beta$ and $\gamma$} \\ + & T^{-1} & \text{the inverse of the linear transformation $T$} \\ +\end{align*} +\begin{align*} + & T^\dag & \text{the pseudoinverse of the linear transformation $T$} \\ + & T^* & \text{the adjoint of the linear operator $T$} \\ + & T_0 & \text{the zero transformation} \\ + & T^t & \text{the transpose of the linear transformation $T$} \\ + & T_\theta & \text{the rotation transformation by $\theta$} \\ + & T_W & \text{the restriction of $T$ to a subspace $W$} \\ + & \ltr{A} & \text{the trace of the matrix $A$} \\ + & V^* & \text{the dual space of the vector space $V$} \\ + & V/W & \text{the quotient space of $V$ modulo $W$} \\ + & W_1 + \dots + W_k & \text{the sum of subspaces $W_1$ through $W_k$} \\ + & \sum_{i=1}^k W_i & \text{the sum of subspaces $W_i$ through $W_k$} \\ + & W_1 \bigoplus W_2 & \text{the direct sum of subspaces $W_1$ and $W_2$} \\ + & W_1 \bigoplus \dots \bigoplus W_k & \text{the direct sum of subspaces $W_1$ through $W_k$} \\ + & \norm{x} & \text{the norm of the vector $\vec{x}$} \\ + & [x]_\beta & \text{the coordinate vector of $x$ relative to $\beta$} \\ + & \langle x, y \rangle & \text{the inner product of $\vec{x}$ and $\vec{y}$} \\ + & \Z_2 & \text{the field consisting of $0$ and $1$} \\ + & \overline{\vec{z}} & \text{the complex conjugate of $\vec{z}$} \\ + & \vec{0} & \text{the zero vector} \\ +\end{align*} diff --git a/chapter-1/bases-and-dimension.tex b/chapter-1/bases-and-dimension.tex new file mode 100644 index 0000000..f68c6e4 --- /dev/null +++ b/chapter-1/bases-and-dimension.tex @@ -0,0 +1 @@ +\section{Bases and Dimension} diff --git a/chapter-1/chapter-1.tex b/chapter-1/chapter-1.tex new file mode 100644 index 0000000..d8e2b1c --- /dev/null +++ b/chapter-1/chapter-1.tex @@ -0,0 +1,8 @@ +\chapter{Vector Spaces} +\subimport{./}{introduction.tex} +\subimport{./}{vector-spaces.tex} +\subimport{./}{subspaces.tex} +\subimport{./}{linear-combinations-and-systems-of-linear-equations.tex} +\subimport{./}{linear-dependence-and-linear-independence.tex} +\subimport{./}{bases-and-dimension.tex} +\subimport{./}{maximal-linearly-independent-subsets.tex} diff --git a/chapter-1/introduction.tex b/chapter-1/introduction.tex new file mode 100644 index 0000000..6d8b74b --- /dev/null +++ b/chapter-1/introduction.tex @@ -0,0 +1 @@ +\section{Introduction} diff --git a/chapter-1/linear-combinations-and-systems-of-linear-equations.tex b/chapter-1/linear-combinations-and-systems-of-linear-equations.tex new file mode 100644 index 0000000..41744eb --- /dev/null +++ b/chapter-1/linear-combinations-and-systems-of-linear-equations.tex @@ -0,0 +1 @@ +\section{Linear Combinations and Systems of Linear Equations} diff --git a/chapter-1/linear-dependence-and-linear-independence.tex b/chapter-1/linear-dependence-and-linear-independence.tex new file mode 100644 index 0000000..69df46a --- /dev/null +++ b/chapter-1/linear-dependence-and-linear-independence.tex @@ -0,0 +1 @@ +\section{Linear Dependence and Linear Independence} diff --git a/chapter-1/maximal-linearly-independent-subsets.tex b/chapter-1/maximal-linearly-independent-subsets.tex new file mode 100644 index 0000000..d37d111 --- /dev/null +++ b/chapter-1/maximal-linearly-independent-subsets.tex @@ -0,0 +1 @@ +\section{Maximal Linearly Independent Subsets} diff --git a/chapter-1/subspaces.tex b/chapter-1/subspaces.tex new file mode 100644 index 0000000..e09e77d --- /dev/null +++ b/chapter-1/subspaces.tex @@ -0,0 +1 @@ +\section{Subspaces} diff --git a/chapter-1/vector-spaces.tex b/chapter-1/vector-spaces.tex new file mode 100644 index 0000000..af0fad4 --- /dev/null +++ b/chapter-1/vector-spaces.tex @@ -0,0 +1 @@ +\section{Vector Spaces} diff --git a/chapter-2/chapter-2.tex b/chapter-2/chapter-2.tex new file mode 100644 index 0000000..1bf2adc --- /dev/null +++ b/chapter-2/chapter-2.tex @@ -0,0 +1,8 @@ +\chapter{Linear Transformations and Matrices} +\subimport{./}{linear-transformations-null-spaces-and-ranges.tex} +\subimport{./}{the-matrix-representation-of-a-linear-transformation.tex} +\subimport{./}{composition-of-linear-transformations-and-matrix-multiplication.tex} +\subimport{./}{invertibility-and-isomorphisms.tex} +\subimport{./}{the-change-of-coordinate-matrix.tex} +\subimport{./}{dual-spaces.tex} +\subimport{./}{homogeneous-linear-differential-equations-with-constant-coefficients.tex} diff --git a/chapter-2/composition-of-linear-transformations-and-matrix-multiplication.tex b/chapter-2/composition-of-linear-transformations-and-matrix-multiplication.tex new file mode 100644 index 0000000..de2b721 --- /dev/null +++ b/chapter-2/composition-of-linear-transformations-and-matrix-multiplication.tex @@ -0,0 +1 @@ +\section{Compositions of Linear Transformations and Matrix Multiplication} diff --git a/chapter-2/dual-spaces.tex b/chapter-2/dual-spaces.tex new file mode 100644 index 0000000..f1e3db8 --- /dev/null +++ b/chapter-2/dual-spaces.tex @@ -0,0 +1 @@ +\section{Dual Spaces} diff --git a/chapter-2/homogeneous-linear-differential-equations-with-constant-coefficients.tex b/chapter-2/homogeneous-linear-differential-equations-with-constant-coefficients.tex new file mode 100644 index 0000000..5ac7777 --- /dev/null +++ b/chapter-2/homogeneous-linear-differential-equations-with-constant-coefficients.tex @@ -0,0 +1 @@ +\section{Homogeneous Linear Differential Equations with Constant Coefficients} diff --git a/chapter-2/invertibility-and-isomorphisms.tex b/chapter-2/invertibility-and-isomorphisms.tex new file mode 100644 index 0000000..635438d --- /dev/null +++ b/chapter-2/invertibility-and-isomorphisms.tex @@ -0,0 +1 @@ +\section{Invertibility and Isomorphisms} diff --git a/chapter-2/linear-transformations-null-spaces-and-ranges.tex b/chapter-2/linear-transformations-null-spaces-and-ranges.tex new file mode 100644 index 0000000..60dd611 --- /dev/null +++ b/chapter-2/linear-transformations-null-spaces-and-ranges.tex @@ -0,0 +1 @@ +\section{Linear Transformations, Null Spaces, and Ranges} diff --git a/chapter-2/the-change-of-coordinate-matrix.tex b/chapter-2/the-change-of-coordinate-matrix.tex new file mode 100644 index 0000000..334c9bf --- /dev/null +++ b/chapter-2/the-change-of-coordinate-matrix.tex @@ -0,0 +1 @@ +\section{The Change of Coordinate Matrix} diff --git a/chapter-2/the-matrix-representation-of-a-linear-transformation.tex b/chapter-2/the-matrix-representation-of-a-linear-transformation.tex new file mode 100644 index 0000000..7d0fb3a --- /dev/null +++ b/chapter-2/the-matrix-representation-of-a-linear-transformation.tex @@ -0,0 +1 @@ +\section{The Matrix Representation of a Linear Transformation} diff --git a/chapter-3/chapter-3.tex b/chapter-3/chapter-3.tex new file mode 100644 index 0000000..5429e78 --- /dev/null +++ b/chapter-3/chapter-3.tex @@ -0,0 +1,5 @@ +\chapter{Elementary Matrix Operations and Systems of Linear Equations} +\subimport{./}{elementary-matrix-operations-and-elementary-matrices.tex} +\subimport{./}{the-rank-of-a-matrix-and-matrix-inverses.tex} +\subimport{./}{systems-of-linear-equations-theoretical-aspects.tex} +\subimport{./}{systems-of-linear-equations-computational-aspects.tex} diff --git a/chapter-3/elementary-matrix-operations-and-elementary-matrices.tex b/chapter-3/elementary-matrix-operations-and-elementary-matrices.tex new file mode 100644 index 0000000..01b78bc --- /dev/null +++ b/chapter-3/elementary-matrix-operations-and-elementary-matrices.tex @@ -0,0 +1 @@ +\section{Elementary Matrix Operations and Elementary Matrices} diff --git a/chapter-3/systems-of-linear-equations-computational-aspects.tex b/chapter-3/systems-of-linear-equations-computational-aspects.tex new file mode 100644 index 0000000..1e06106 --- /dev/null +++ b/chapter-3/systems-of-linear-equations-computational-aspects.tex @@ -0,0 +1 @@ +\section{Systems of Linear Equations -- Computational Aspects} diff --git a/chapter-3/systems-of-linear-equations-theoretical-aspects.tex b/chapter-3/systems-of-linear-equations-theoretical-aspects.tex new file mode 100644 index 0000000..97e5b6a --- /dev/null +++ b/chapter-3/systems-of-linear-equations-theoretical-aspects.tex @@ -0,0 +1 @@ +\section{Systems of Linear Equations -- Theoretical Aspects} diff --git a/chapter-3/the-rank-of-a-matrix-and-matrix-inverses.tex b/chapter-3/the-rank-of-a-matrix-and-matrix-inverses.tex new file mode 100644 index 0000000..767e1a9 --- /dev/null +++ b/chapter-3/the-rank-of-a-matrix-and-matrix-inverses.tex @@ -0,0 +1 @@ +\section{The Rank of a Matrix and Matrix Inverses} diff --git a/chapter-4/a-characterization-of-the-determinant.tex b/chapter-4/a-characterization-of-the-determinant.tex new file mode 100644 index 0000000..0099b74 --- /dev/null +++ b/chapter-4/a-characterization-of-the-determinant.tex @@ -0,0 +1 @@ +\section{A Characterization of the Determinant} diff --git a/chapter-4/chapter-4.tex b/chapter-4/chapter-4.tex new file mode 100644 index 0000000..d1a7655 --- /dev/null +++ b/chapter-4/chapter-4.tex @@ -0,0 +1,6 @@ +\chapter{Determinants} +\subimport{./}{determinants-of-order-2.tex} +\subimport{./}{determinants-of-order-n.tex} +\subimport{./}{properties-of-determinants.tex} +\subimport{./}{summary-important-facts-about-determinants.tex} +\subimport{./}{a-characterization-of-the-determinant.tex} diff --git a/chapter-4/determinants-of-order-2.tex b/chapter-4/determinants-of-order-2.tex new file mode 100644 index 0000000..e7599e9 --- /dev/null +++ b/chapter-4/determinants-of-order-2.tex @@ -0,0 +1 @@ +\section{Determinants of Order 2} diff --git a/chapter-4/determinants-of-order-n.tex b/chapter-4/determinants-of-order-n.tex new file mode 100644 index 0000000..2b93fa1 --- /dev/null +++ b/chapter-4/determinants-of-order-n.tex @@ -0,0 +1 @@ +\section{Determinants of Order $n$} diff --git a/chapter-4/properties-of-determinants.tex b/chapter-4/properties-of-determinants.tex new file mode 100644 index 0000000..a18dc6e --- /dev/null +++ b/chapter-4/properties-of-determinants.tex @@ -0,0 +1 @@ +\section{Properties of Determinants} diff --git a/chapter-4/summary-important-facts-about-determinants.tex b/chapter-4/summary-important-facts-about-determinants.tex new file mode 100644 index 0000000..f9ed462 --- /dev/null +++ b/chapter-4/summary-important-facts-about-determinants.tex @@ -0,0 +1 @@ +\section{Summary -- Important Facts about Determinants} diff --git a/chapter-5/chapter-5.tex b/chapter-5/chapter-5.tex new file mode 100644 index 0000000..eaff4cb --- /dev/null +++ b/chapter-5/chapter-5.tex @@ -0,0 +1,5 @@ +\chapter{Diagonalization} +\subimport{./}{eigenvalues-and-eigenvectors.tex} +\subimport{./}{diagonalizability.tex} +\subimport{./}{matrix-limits-and-markov-chains.tex} +\subimport{./}{invariant-subspaces-and-the-cayley-hamilton-theorem.tex} diff --git a/chapter-5/diagonalizability.tex b/chapter-5/diagonalizability.tex new file mode 100644 index 0000000..920bdd6 --- /dev/null +++ b/chapter-5/diagonalizability.tex @@ -0,0 +1 @@ +\section{Diagonalizability} diff --git a/chapter-5/eigenvalues-and-eigenvectors.tex b/chapter-5/eigenvalues-and-eigenvectors.tex new file mode 100644 index 0000000..1c5b62c --- /dev/null +++ b/chapter-5/eigenvalues-and-eigenvectors.tex @@ -0,0 +1 @@ +\section{Eigenvalues and Eigenvectors} diff --git a/chapter-5/invariant-subspaces-and-the-cayley-hamilton-theorem.tex b/chapter-5/invariant-subspaces-and-the-cayley-hamilton-theorem.tex new file mode 100644 index 0000000..f4c3dec --- /dev/null +++ b/chapter-5/invariant-subspaces-and-the-cayley-hamilton-theorem.tex @@ -0,0 +1 @@ +\section{Invariant Subspaces and the Cayley-Hamilton Theorem} diff --git a/chapter-5/matrix-limits-and-markov-chains.tex b/chapter-5/matrix-limits-and-markov-chains.tex new file mode 100644 index 0000000..e937ba1 --- /dev/null +++ b/chapter-5/matrix-limits-and-markov-chains.tex @@ -0,0 +1 @@ +\section{Matrix Limits and Markov Chains} diff --git a/chapter-6/bilinear-and-quadratic-forms.tex b/chapter-6/bilinear-and-quadratic-forms.tex new file mode 100644 index 0000000..1cb208a --- /dev/null +++ b/chapter-6/bilinear-and-quadratic-forms.tex @@ -0,0 +1 @@ +\section{Bilinear and Quadratic Forms} diff --git a/chapter-6/chapter-6.tex b/chapter-6/chapter-6.tex new file mode 100644 index 0000000..b897375 --- /dev/null +++ b/chapter-6/chapter-6.tex @@ -0,0 +1,12 @@ +\chapter{Inner Product Spaces} +\subimport{./}{inner-products-and-norms.tex} +\subimport{./}{the-gram-schmidt-orthogonalization-process-and-orthogonal-complements.tex} +\subimport{./}{the-adjoint-of-a-linear-operator.tex} +\subimport{./}{normal-and-self-adjoint-operators.tex} +\subimport{./}{unitary-and-orthogonal-operators-and-their-matrices.tex} +\subimport{./}{orthogonal-projections-and-the-spectral-theorem.tex} +\subimport{./}{the-singular-value-decomposition-and-the-pseudoinverse.tex} +\subimport{./}{bilinear-and-quadratic-forms.tex} +\subimport{./}{einsteins-special-theory-of-relativity.tex} +\subimport{./}{conditioning-and-the-rayleigh-quotient.tex} +\subimport{./}{the-geometry-of-orthogonal-operators.tex} diff --git a/chapter-6/conditioning-and-the-rayleigh-quotient.tex b/chapter-6/conditioning-and-the-rayleigh-quotient.tex new file mode 100644 index 0000000..b48c692 --- /dev/null +++ b/chapter-6/conditioning-and-the-rayleigh-quotient.tex @@ -0,0 +1 @@ +\section{Conditioning and the Rayleigh Quotient} diff --git a/chapter-6/einsteins-special-theory-of-relativity.tex b/chapter-6/einsteins-special-theory-of-relativity.tex new file mode 100644 index 0000000..fb26195 --- /dev/null +++ b/chapter-6/einsteins-special-theory-of-relativity.tex @@ -0,0 +1 @@ +\section{Einstein's Special Theory of Relativity} diff --git a/chapter-6/inner-products-and-norms.tex b/chapter-6/inner-products-and-norms.tex new file mode 100644 index 0000000..f2bc235 --- /dev/null +++ b/chapter-6/inner-products-and-norms.tex @@ -0,0 +1 @@ +\section{Inner Products and Norms} diff --git a/chapter-6/normal-and-self-adjoint-operators.tex b/chapter-6/normal-and-self-adjoint-operators.tex new file mode 100644 index 0000000..c2f3a67 --- /dev/null +++ b/chapter-6/normal-and-self-adjoint-operators.tex @@ -0,0 +1 @@ +\section{Normal and Self-Adjoing Operators} diff --git a/chapter-6/orthogonal-projections-and-the-spectral-theorem.tex b/chapter-6/orthogonal-projections-and-the-spectral-theorem.tex new file mode 100644 index 0000000..c937a5d --- /dev/null +++ b/chapter-6/orthogonal-projections-and-the-spectral-theorem.tex @@ -0,0 +1 @@ +\section{Orthogonal Projections and the Spectral Theorem} diff --git a/chapter-6/the-adjoint-of-a-linear-operator.tex b/chapter-6/the-adjoint-of-a-linear-operator.tex new file mode 100644 index 0000000..0ab2a58 --- /dev/null +++ b/chapter-6/the-adjoint-of-a-linear-operator.tex @@ -0,0 +1 @@ +\section{The Adjoint of a Linear Operator} diff --git a/chapter-6/the-geometry-of-orthogonal-operators.tex b/chapter-6/the-geometry-of-orthogonal-operators.tex new file mode 100644 index 0000000..e907493 --- /dev/null +++ b/chapter-6/the-geometry-of-orthogonal-operators.tex @@ -0,0 +1 @@ +\section{The Geometry of Orthogonal Operators} diff --git a/chapter-6/the-gram-schmidt-orthogonalization-process-and-orthogonal-complements.tex b/chapter-6/the-gram-schmidt-orthogonalization-process-and-orthogonal-complements.tex new file mode 100644 index 0000000..3730b2d --- /dev/null +++ b/chapter-6/the-gram-schmidt-orthogonalization-process-and-orthogonal-complements.tex @@ -0,0 +1 @@ +\section{The Gram-Schmidt Orthogonalization Process and Orthogonal Complements} diff --git a/chapter-6/the-singular-value-decomposition-and-the-pseudoinverse.tex b/chapter-6/the-singular-value-decomposition-and-the-pseudoinverse.tex new file mode 100644 index 0000000..b2afe54 --- /dev/null +++ b/chapter-6/the-singular-value-decomposition-and-the-pseudoinverse.tex @@ -0,0 +1 @@ +\section{The Singular Value Decomposition and the Pseudoinverse} diff --git a/chapter-6/unitary-and-orthogonal-operators-and-their-matrices.tex b/chapter-6/unitary-and-orthogonal-operators-and-their-matrices.tex new file mode 100644 index 0000000..540692f --- /dev/null +++ b/chapter-6/unitary-and-orthogonal-operators-and-their-matrices.tex @@ -0,0 +1 @@ +\section{Unitary and Orthogonal Operators and Their Matrices} diff --git a/chapter-7/chapter-7.tex b/chapter-7/chapter-7.tex new file mode 100644 index 0000000..7905705 --- /dev/null +++ b/chapter-7/chapter-7.tex @@ -0,0 +1,5 @@ +\chapter{Canonical Forms} +\subimport{./}{the-jordan-canonical-form-i.tex} +\subimport{./}{the-jordan-canonical-form-ii.tex} +\subimport{./}{the-minimal-polynomial.tex} +\subimport{./}{the-rational-canonical-form.tex} diff --git a/chapter-7/the-jordan-canonical-form-i.tex b/chapter-7/the-jordan-canonical-form-i.tex new file mode 100644 index 0000000..be389e3 --- /dev/null +++ b/chapter-7/the-jordan-canonical-form-i.tex @@ -0,0 +1 @@ +\section{The Jordan Canonical Form I} diff --git a/chapter-7/the-jordan-canonical-form-ii.tex b/chapter-7/the-jordan-canonical-form-ii.tex new file mode 100644 index 0000000..0bfba8f --- /dev/null +++ b/chapter-7/the-jordan-canonical-form-ii.tex @@ -0,0 +1 @@ +\section{The Jordan Canonical Form II} diff --git a/chapter-7/the-minimal-polynomial.tex b/chapter-7/the-minimal-polynomial.tex new file mode 100644 index 0000000..1f861ae --- /dev/null +++ b/chapter-7/the-minimal-polynomial.tex @@ -0,0 +1 @@ +\section{The Minimal Polynomial} diff --git a/chapter-7/the-rational-canonical-form.tex b/chapter-7/the-rational-canonical-form.tex new file mode 100644 index 0000000..6b8cb18 --- /dev/null +++ b/chapter-7/the-rational-canonical-form.tex @@ -0,0 +1 @@ +\section{The rational Canonical Form} diff --git a/init.sty b/init.sty new file mode 100644 index 0000000..d4dd92c --- /dev/null +++ b/init.sty @@ -0,0 +1,59 @@ +\ProvidesPackage{init} + +\usepackage{import} +\usepackage[utf8]{inputenc} +\usepackage{pgfplots} +\usepackage[english]{babel} +\usepackage{amsthm} +\usepackage{thmtools} +\usepackage{hyperref} +\usepackage{cancel} +\usepackage{mathtools} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{graphicx} +\usepackage{relsize} +\usepackage{listings} +\graphicspath{ {./images/} } +\usepackage{array} +\usepackage{tikz} +\usetikzlibrary{arrows} +\usepackage[left=2cm, right=2.5cm, top=2.5cm, bottom=2.5cm]{geometry} +\usepackage{enumitem} +\usepackage{mathrsfs} + +% Math Functions +\newcommand{\limx}[2]{\displaystyle\lim\limits_{#1 \to #2}} +\newcommand{\st}{\ \text{s.t.}\ } +\newcommand{\abs}[1]{\left\lvert #1 \right\rvert} +\newcommand{\dotp}{\dot{\mathcal{P}}} +\newcommand{\dotq}{\dot{\mathcal{Q}}} +\newcommand{\Int}[1]{\text{int}\left(#1\right)} +\newcommand{\cl}[1]{\text{cl}\left(#1\right)} +\newcommand{\bd}[1]{\text{bd}\left(#1\right)} +\newcommand{\lr}[1]{\left(#1\right)} +\newcommand{\lspan}[1]{\text{span}\left(#1\right)} +\newcommand{\ldim}[1]{\text{dim}\left(#1\right)} +\newcommand{\nullity}[1]{\text{nullity}\left(#1\right)} +\newcommand{\rank}[1]{\text{rank}\left(#1\right)} +\newcommand{\ldet}[1]{\text{det}\left(#1\right)} +\newcommand{\ltr}[1]{\text{tr}\left(#1\right)} +\newcommand{\norm}[1]{\left\lVert#1\right\rVert} +\DeclareMathOperator{\sign}{sgn} +\renewcommand{\qedsymbol}{$\blacksquare$} + +% Special Sets +\newcommand{\R}{\mathbb{R}} +\newcommand{\N}{\mathbb{N}} +\newcommand{\Q}{\mathbb{Q}} +\newcommand{\C}{\mathbb{C}} +\newcommand{\Z}{\mathbb{Z}} +\newcommand{\F}{\mathbb{F}} + +% Theorem Styles +\declaretheorem[numberwithin=subsection, style=definition]{theorem, definition, notation, lemma, corollary, remark, example} + +% Formatting +\setlist[enumerate]{font=\bfseries} +