Skip to main content\(\require{cancel}\newcommand{\Z}{{\mathbb Z}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\C}{{\mathbb C}}
\newcommand{\T}{{\mathbb T}}
\newcommand{\F}{{\mathbb F}}
\newcommand{\HH}{{\mathbb H}}
\newcommand{\compose}{\circ}
\newcommand{\bolda}{{\mathbf a}}
\newcommand{\boldb}{{\mathbf b}}
\newcommand{\boldc}{{\mathbf c}}
\newcommand{\boldd}{{\mathbf d}}
\newcommand{\bolde}{{\mathbf e}}
\newcommand{\boldi}{{\mathbf i}}
\newcommand{\boldj}{{\mathbf j}}
\newcommand{\boldk}{{\mathbf k}}
\newcommand{\boldn}{{\mathbf n}}
\newcommand{\boldp}{{\mathbf p}}
\newcommand{\boldq}{{\mathbf q}}
\newcommand{\boldr}{{\mathbf r}}
\newcommand{\bolds}{{\mathbf s}}
\newcommand{\boldt}{{\mathbf t}}
\newcommand{\boldu}{{\mathbf u}}
\newcommand{\boldv}{{\mathbf v}}
\newcommand{\boldw}{{\mathbf w}}
\newcommand{\boldx}{{\mathbf x}}
\newcommand{\boldy}{{\mathbf y}}
\newcommand{\boldz}{{\mathbf z}}
\newcommand{\boldzero}{{\mathbf 0}}
\newcommand{\boldmod}{\boldsymbol{ \bmod }}
\newcommand{\boldT}{{\mathbf T}}
\newcommand{\boldN}{{\mathbf N}}
\newcommand{\boldB}{{\mathbf B}}
\newcommand{\boldF}{{\mathbf F}}
\newcommand{\boldS}{{\mathbf S}}
\newcommand{\boldG}{{\mathbf G}}
\newcommand{\boldK}{{\mathbf K}}
\newcommand{\boldL}{{\mathbf L}}
\DeclareMathOperator{\lcm}{lcm}
\DeclareMathOperator{\Span}{span}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\NS}{null}
\DeclareMathOperator{\RS}{row}
\DeclareMathOperator{\CS}{col}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\range}{range}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\nullity}{nullity}
\DeclareMathOperator{\sign}{sign}
\DeclareMathOperator{\Fix}{Fix}
\DeclareMathOperator{\Aff}{Aff}
\DeclareMathOperator{\Frac}{Frac}
\DeclareMathOperator{\Ann}{Ann}
\DeclareMathOperator{\Tor}{Tor}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\mdeg}{mdeg}
\DeclareMathOperator{\Lt}{Lt}
\DeclareMathOperator{\Lc}{Lc}
\DeclareMathOperator{\disc}{disc}
\DeclareMathOperator{\Frob}{Frob}
\DeclareMathOperator{\adj}{adj}
\DeclareMathOperator{\curl}{curl}
\DeclareMathOperator{\grad}{grad}
\DeclareMathOperator{\diver}{div}
\DeclareMathOperator{\flux}{flux}
\def\Gal{\operatorname{Gal}}
\def\ord{\operatorname{ord}}
\def\ML{\operatorname{M}}
\def\GL{\operatorname{GL}}
\def\PGL{\operatorname{PGL}}
\def\SL{\operatorname{SL}}
\def\PSL{\operatorname{PSL}}
\def\GSp{\operatorname{GSp}}
\def\PGSp{\operatorname{PGSp}}
\def\Sp{\operatorname{Sp}}
\def\PSp{\operatorname{PSp}}
\def\Aut{\operatorname{Aut}}
\def\Inn{\operatorname{Inn}}
\def\Hom{\operatorname{Hom}}
\def\End{\operatorname{End}}
\def\ch{\operatorname{char}}
\def\Zp{\Z/p\Z}
\def\Zm{\Z/m\Z}
\def\Zn{\Z/n\Z}
\def\Fp{\F_p}
\newcommand{\surjects}{\twoheadrightarrow}
\newcommand{\injects}{\hookrightarrow}
\newcommand{\bijects}{\leftrightarrow}
\newcommand{\isomto}{\overset{\sim}{\rightarrow}}
\newcommand{\floor}[1]{\lfloor#1\rfloor}
\newcommand{\ceiling}[1]{\left\lceil#1\right\rceil}
\newcommand{\mclass}[2][m]{[#2]_{#1}}
\newcommand{\val}[2][]{\left\lvert #2\right\rvert_{#1}}
\newcommand{\abs}[2][]{\left\lvert #2\right\rvert_{#1}}
\newcommand{\valuation}[2][]{\left\lvert #2\right\rvert_{#1}}
\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
\newcommand{\anpoly}{a_nx^n+a_{n-1}x^{n-1}\cdots +a_1x+a_0}
\newcommand{\anmonic}{x^n+a_{n-1}x^{n-1}\cdots +a_1x+a_0}
\newcommand{\bmpoly}{b_mx^m+b_{m-1}x^{m-1}\cdots +b_1x+b_0}
\newcommand{\bnpoly}{b_nx^n+b_{n-1}x^{n-1}\cdots +b_1x+b_0}
\newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}
\newcommand{\normalin}{\trianglelefteq}
\newcommand{\angvec}[1]{\langle #1\rangle}
\newcommand{\varpoly}[2]{#1_{#2}x^{#2}+#1_{#2-1}x^{#2-1}\cdots +#1_1x+#1_0}
\newcommand{\varpower}[1][a]{#1_0+#1_1x+#1_1x^2+\cdots}
\newcommand{\limasto}[2][x]{\lim_{#1\rightarrow #2}}
\def\ntoinfty{\lim_{n\rightarrow\infty}}
\def\xtoinfty{\lim_{x\rightarrow\infty}}
\def\ii{\item}
\def\bb{\begin{enumerate}}
\def\ee{\end{enumerate}}
\def\ds{\displaystyle}
\def\p{\partial}
\newcommand{\abcdmatrix}[4]{\begin{bmatrix}#1\amp #2\\ #3\amp #4 \end{bmatrix}
}
\newenvironment{amatrix}[1][ccc|c]{\left[\begin{array}{#1}}{\end{array}\right]}
\newenvironment{linsys}[2][m]{
\begin{array}[#1]{@{}*{#2}{rc}r@{}}
}{
\end{array}}
\newcommand{\eqsys}{\begin{array}{rcrcrcr}
a_{11}x_{1}\amp +\amp a_{12}x_{2}\amp +\cdots+\amp a_{1n}x_{n}\amp =\amp b_1\\
a_{21}x_{1}\amp +\amp a_{22}x_{2}\amp +\cdots+\amp a_{2n}x_{n}\amp =\amp b_2\\
\amp \vdots\amp \amp \vdots \amp \amp \vdots \amp \\
a_{m1}x_{1}\amp +\amp a_{m2}x_{2}\amp +\cdots +\amp a_{mn}x_{n}\amp =\amp b_m
\end{array}
}
\newcommand{\numeqsys}{\begin{array}{rrcrcrcr}
e_1:\amp a_{11}x_{1}\amp +\amp a_{12}x_{2}\amp +\cdots+\amp a_{1n}x_{n}\amp =\amp b_1\\
e_2: \amp a_{21}x_{1}\amp +\amp a_{22}x_{2}\amp +\cdots+\amp a_{2n}x_{n}\amp =\amp b_2\\
\amp \vdots\amp \amp \vdots \amp \amp \vdots \amp \\
e_m: \amp a_{m1}x_{1}\amp +\amp a_{m2}x_{2}\amp +\cdots +\amp a_{mn}x_{n}\amp =\amp b_m
\end{array}
}
\newcommand{\homsys}{\begin{array}{rcrcrcr}
a_{11}x_{1}\amp +\amp a_{12}x_{2}\amp +\cdots+\amp a_{1n}x_{n}\amp =\amp 0\\
a_{21}x_{1}\amp +\amp a_{22}x_{2}\amp +\cdots+\amp a_{2n}x_{n}\amp =\amp 0\\
\amp \vdots\amp \amp \vdots \amp \amp \vdots \amp \\
a_{m1}x_{1}\amp +\amp a_{m2}x_{2}\amp +\cdots +\amp a_{mn}x_{n}\amp =\amp 0
\end{array}
}
\newcommand{\vareqsys}[4]{
\begin{array}{ccccccc}
#3_{11}x_{1}\amp +\amp #3_{12}x_{2}\amp +\cdots+\amp #3_{1#2}x_{#2}\amp =\amp #4_1\\
#3_{21}x_{1}\amp +\amp #3_{22}x_{2}\amp +\cdots+\amp #3_{2#2}x_{#2}\amp =\amp #4_2\\
\vdots \amp \amp \vdots \amp \amp \vdots \amp =\amp \\
#3_{#1 1}x_{1}\amp +\amp #3_{#1 2}x_{2}\amp +\cdots +\amp #3_{#1 #2}x_{#2}\amp =\amp #4_{#1}
\end{array}
}
\newcommand{\genmatrix}[1][a]{
\begin{bmatrix}
#1_{11} \amp #1_{12} \amp \cdots \amp #1_{1n} \\
#1_{21} \amp #1_{22} \amp \cdots \amp #1_{2n} \\
\vdots \amp \vdots \amp \ddots \amp \vdots \\
#1_{m1} \amp #1_{m2} \amp \cdots \amp #1_{mn}
\end{bmatrix}
}
\newcommand{\varmatrix}[3]{
\begin{bmatrix}
#3_{11} \amp #3_{12} \amp \cdots \amp #3_{1#2} \\
#3_{21} \amp #3_{22} \amp \cdots \amp #3_{2#2} \\
\vdots \amp \vdots \amp \ddots \amp \vdots \\
#3_{#1 1} \amp #3_{#1 2} \amp \cdots \amp #3_{#1 #2}
\end{bmatrix}
}
\newcommand{\augmatrix}{
\begin{amatrix}[cccc|c]
a_{11} \amp a_{12} \amp \cdots \amp a_{1n} \amp b_{1}\\
a_{21} \amp a_{22} \amp \cdots \amp a_{2n} \amp b_{2}\\
\vdots \amp \vdots \amp \ddots \amp \vdots \amp \vdots\\
a_{m1} \amp a_{m2} \amp \cdots \amp a_{mn}\amp b_{m}
\end{amatrix}
}
\newcommand{\varaugmatrix}[4]{
\begin{amatrix}[cccc|c]
#3_{11} \amp #3_{12} \amp \cdots \amp #3_{1#2} \amp #4_{1}\\
#3_{21} \amp #3_{22} \amp \cdots \amp #3_{2#2} \amp #4_{2}\\
\vdots \amp \vdots \amp \ddots \amp \vdots \amp \vdots\\
#3_{#1 1} \amp #3_{#1 2} \amp \cdots \amp #3_{#1 #2}\amp #4_{#1}
\end{amatrix}
}
\newcommand{\spaceforemptycolumn}{\makebox[\wd\boxofmathplus]{\ }}
\newcommand{\generalmatrix}[3]{
\left(
\begin{array}{cccc}
#1_{1,1} \amp #1_{1,2} \amp \ldots \amp #1_{1,#2} \\
#1_{2,1} \amp #1_{2,2} \amp \ldots \amp #1_{2,#2} \\
\amp \vdots \\
#1_{#3,1} \amp #1_{#3,2} \amp \ldots \amp #1_{#3,#2}
\end{array}
\right) }
\newcommand{\colvec}[2][c]{\begin{amatrix}[#1] #2 \end{amatrix}}
\newcommand{\rowvec}[1]{\begin{bmatrix} #1 \end{bmatrix}}
\DeclareMathOperator{\size}{size}
\DeclareMathOperator{\adjoint}{adj}
\DeclareMathOperator{\sgn}{sgn}
\newcommand{\restrictionmap}[2]{{#1}\mathpunct\upharpoonright\hbox{}_{#2}}
\renewcommand{\emptyset}{\varnothing}
\newcommand{\proj}[2]{\mbox{proj}_{#2}({#1}) }
\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}
\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\,}$}}}
\)
Appendix G Figures and video examples
0.5 Proof techniques
Figure 0.5.9 Mathematical induction as ladder of propositions
1.1 Systems of linear equations
Figure 1.1.8 Using Sage to visualize
\(\mathcal{P}\colon ax+by+cz=d\) via normal vector
\(\boldn=(a,b,c)\)
1.3 Solving linear systems
2.1 Matrix arithmetic
2.2 Matrix algebra
2.4 The invertibility theorem
2.5 The determinant
3.1 Real vector spaces
3.2 Linear transformations
3.3 Subspaces
Figure 3.3.5 Video: deciding if
\(W\subseteq V\) is a subspace
Figure 3.3.6 Video: deciding if
\(W\subseteq V\) is a subspace
3.4 Null space and image
3.5 Span and linear independence
3.6 Bases
3.7 Dimension
3.8 Rank-nullity theorem and fundamental spaces
4.1 Coordinate vectors
4.2 Matrix representations of linear transformations
4.3 Change of basis
4.4 Eigenvectors and eigenvalues
4.5 Diagonalization
5.3 Orthogonal projection
5.4 The spectral theorem
