\renewcommand{\(}{\left(} \renewcommand{\)}{\right)} \renewcommand{\i}{^{-1}} \newcommand{\mathnum}{\mathbb} \newcommand{\mathnumsbold}{\renewcommand{\mathnum}{\mathbf}} \newcommand{\A}{\mathbf{A}} \newcommand{\B}{\mathcal{B}} \newcommand{\C}{\mathnum{C}} \newcommand{\D}{\mathbb{D}} \newcommand{\E}{\mathbb{E}} \newcommand{\F}{\mathnum{F}} \renewcommand{\H}{\mathbf{H}} \newcommand{\I}{\mathbf{I}} \newcommand{\M}{\mathbb{M}} \newcommand{\N}{\mathnum{N}} \renewcommand{\P}{\mathbb{P}} \newcommand{\Q}{\mathnum{Q}} \newcommand{\R}{\mathnum{R}} \renewcommand{\S}{\mathcal{S}} \newcommand{\U}{\mathbf{U}} \newcommand{\Z}{\mathnum{Z}} \renewcommand{\phi}{\varphi} \newcommand{\mathvec}[1]{\mathbf{#1}} \renewcommand{\a}{\mathvec{a}} \renewcommand{\b}{\mathvec{b}} \renewcommand{\c}{\mathvec{c}} \newcommand{\e}{\mathbf{e}} \newcommand{\f}{\mathbf{f}} \newcommand{\g}{\mathbf{g}} \renewcommand{\r}{\mathvec{r}} \renewcommand{\u}{\mathvec{u}} \renewcommand{\v}{\mathvec{v}} \newcommand{\w}{\mathvec{w}} \newcommand{\x}{\mathvec{x}} \newcommand{\y}{\mathvec{y}} \newcommand{\z}{\mathvec{z}} \newcommand{\0}{\mathvec{0}} \newcommand{\ivec}{\mathvec{i}} \newcommand{\jvec}{\mathvec{j}} \newcommand{\kvec}{\mathvec{k}} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\Col}{Col} \DeclareMathOperator{\Null}{Null} \DeclareMathOperator{\Row}{Row} \DeclareMathOperator{\Spanname}{span} \DeclareMathOperator{\adj}{adj} \DeclareMathOperator{\characteristic}{char} \DeclareMathOperator{\charpoly}{char~poly} \DeclareMathOperator{\cod}{cod} \DeclareMathOperator{\curl}{curl} \DeclareMathOperator{\divergence}{div} \DeclareMathOperator{\dom}{dom} \DeclareMathOperator{\even}{even} \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\GF}{GF} \DeclareMathOperator{\grint}{int} \DeclareMathOperator{\grad}{grad} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\Inn}{Inn} \DeclareMathOperator{\lcm}{lcm} \DeclareMathOperator{\LCM}{LCM} \DeclareMathOperator{\mult}{multiplicity} \DeclareMathOperator{\nullity}{nullity} \DeclareMathOperator{\odd}{odd} \DeclareMathOperator{\orb}{orb} \DeclareMathOperator{\proj}{proj} \DeclareMathOperator{\range}{ran} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\rref}{rref} \DeclareMathOperator{\stab}{stab} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\sinc}{sinc} \DeclareMathOperator{\Sym}{Sym} \DeclareMathOperator{\tr}{tr} \newcommand{\abs}[1]{\left\lvert #1\right\rvert} \renewcommand{\bar}[1]{\overline{#1}} \newcommand{\birestrict}[3]{\left.#1\right\rvert_{#2}^{#3}} %\newcommand{\dbinom}{\displaystyle\binom} \newcommand{\ddt}{\deriv{}{t}} \newcommand{\ddx}{\deriv{}{x}} \newcommand{\deriv}[2]{\dfrac{d#1}{d#2}} \newcommand{\dint}{\displaystyle\int} \newcommand{\dlim}{\displaystyle\lim} \newcommand{\doubleint}[1]{\underset{#1}{\dint\!\!\!\!\dint}} \newcommand{\dstirling}{\displaystyle\stirling} \newcommand{\dsum}{\displaystyle\sum} \newcommand{\gen}[1]{\left\langle#1\right\rangle} \newcommand{\half}{\frac{1}{2}} \newcommand{\hpm}{\hphantom{-}} \newcommand{\ip}[2]{\left\langle{#1},{#2}\right\rangle} \newcommand{\isom}{\simeq} \newcommand{\legendre}[2]{\(\dfrac{#1}{#2}\)} \newcommand{\len}[1]{\left\lVert#1\right\rVert} \newcommand{\marsdendd}[3]{\left.\dfrac{d}{dt}#1(#2+t#3)\right\rvert_{t=0}} \newcommand{\nlim}{\dlim_{n\rightarrow\infty}} \newcommand{\norm}[1]{\left\lVert#1\right\rVert} \newcommand{\pderiv}[2]{\dfrac{\partial{#1}}{\partial{#2}}} \newcommand{\pderivtwo}[2]{\dfrac{\partial^2{#1}}{\partial{#2}^2}} \newcommand{\powerset}[1]{\mathcal{P}(#1)} \newcommand{\restrict}[2]{\left.#1\right\rvert_{#2}} \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\sfrac}[2]{\frac{\mathstrut #1}{\mathstrut #2}} \newcommand{\size}[1]{\left\lvert#1\right\rvert} \newcommand{\Span}[1]{\Spanname\left\{#1\right\}} \newcommand{\stirling}{\genfrac\{\}{0pt}{}} \newcommand{\tm}[1]{$\text{#1}^{\text{TM}}$} \newcommand{\trans}[1]{#1^t} \newcommand{\tripleint}[1]{\underset{#1}{\dint\!\!\!\!\dint\!\!\!\!\dint}} \newcommand{\zero}{\mathbf{0}} \newenvironment{formatrix}[1]% {\setlength{\arraycolsep}{3pt}\left[\hskip -\arraycolsep\begin{array}{#1}}% {\end{array}\hskip -\arraycolsep\right]} \newenvironment{detmatrix}[1]% {\setlength{\arraycolsep}{3pt}\left\lvert\hskip -\arraycolsep\begin{array}{#1}}% {\end{array}\hskip -\arraycolsep\right\rvert} \newcommand{\col}[1]{\begin{formatrix}{r}#1\end{formatrix}} \newcommand{\ccol}[1]{\begin{formatrix}{c}#1\end{formatrix}} \newcommand{\twomat}[4]{\begin{formatrix}{rr}#1\\#3\end{formatrix}} \newcommand{\answerbox}[2]{\framebox[#1][l]{\rule[-1.5ex]{0em}{2.5em}#2}}