Colophon

\(\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}} \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\,}$}}} \)

Colophon Colophon

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Linear algebra: the theory of vector spaces and linear transformations

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Colophon Colophon