Skip to main content\(\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\,}$}}}
\)
Chapter 7 Inner product spaces
