\(\newcommand{\avec}{{\vec{a}}}
\newcommand{\bvec}{{\vec{b}}}
\newcommand{\cvec}{{\vec{c}}}
\newcommand{\dvec}{{\vec{d}}}
\newcommand{\dtil}{\widetilde{\mathbf d}}
\newcommand{\evec}{{\vec{e}}}
\newcommand{\fvec}{{\vec{f}}}
\newcommand{\nvec}{{\vec{n}}}
\newcommand{\pvec}{{\vec{p}}}
\newcommand{\qvec}{{\vec{q}}}
\newcommand{\svec}{{\vec{s}}}
\newcommand{\tvec}{{\vec{t}}}
\newcommand{\uvec}{{\vec{u}}}
\newcommand{\vvec}{{\vec{v}}}
\newcommand{\wvec}{{\vec{w}}}
\newcommand{\xvec}{{\vec{x}}}
\newcommand{\yvec}{{\vec{y}}}
\newcommand{\zvec}{{\vec{z}}}
\newcommand{\rvec}{{\vec{r}}}
\newcommand{\mvec}{{\vec{m}}}
\newcommand{\zerovec}{{\vec{0}}}
\newcommand{\onevec}{{\vec{1}}}
\newcommand{\real}{{\mathbb R}}
\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2
\end{array}\right]}
\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2
\end{array}\right]}
\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3
\end{array}\right]}
\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3
\end{array}\right]}
\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4
\end{array}\right]}
\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4
\end{array}\right]}
\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\
#4 \\ #5 \\ \end{array}\right]}
\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\
#4 \\ #5 \\ \end{array}\right]}
\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}
\newcommand{\laspan}[1]{\text{Span}\{#1\}}
\newcommand{\bcal}{{\cal B}}
\newcommand{\ccal}{{\cal C}}
\newcommand{\scal}{{\cal S}}
\newcommand{\wcal}{{\cal W}}
\newcommand{\ecal}{{\cal E}}
\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}
\newcommand{\gray}[1]{\color{gray}{#1}}
\newcommand{\lgray}[1]{\color{lightgray}{#1}}
\newcommand{\rank}{\operatorname{rank}}
\newcommand{\row}{\text{Row}}
\newcommand{\col}{\text{Col}}
\renewcommand{\row}{\text{Row}}
\newcommand{\nul}{\text{Nul}}
\newcommand{\var}{\text{Var}}
\newcommand{\corr}{\text{corr}}
\newcommand{\len}[1]{\left|#1\right|}
\newcommand{\bbar}{\overline{\bvec}}
\newcommand{\bhat}{\widehat{\bvec}}
\newcommand{\bperp}{\bvec^\perp}
\newcommand{\xhat}{\widehat{\xvec}}
\newcommand{\vhat}{\widehat{\vvec}}
\newcommand{\uhat}{\widehat{\uvec}}
\newcommand{\what}{\widehat{\wvec}}
\newcommand{\Sighat}{\widehat{\Sigma}}
\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\,}$}}}
\)