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\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } 

\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}

\newcommand{\id}{\mathrm{id}} \newcommand{\Span}{\mathrm{span}}

( \newcommand{\kernel}{\mathrm{null}\,}\) \newcommand{\range}{\mathrm{range}\,}

\newcommand{\RealPart}{\mathrm{Re}} \newcommand{\ImaginaryPart}{\mathrm{Im}}

\newcommand{\Argument}{\mathrm{Arg}} \newcommand{\norm}[1]{\| #1 \|}

\newcommand{\inner}[2]{\langle #1, #2 \rangle}

\newcommand{\Span}{\mathrm{span}}

\newcommand{\id}{\mathrm{id}}

\newcommand{\Span}{\mathrm{span}}

\newcommand{\kernel}{\mathrm{null}\,}

\newcommand{\range}{\mathrm{range}\,}

\newcommand{\RealPart}{\mathrm{Re}}

\newcommand{\ImaginaryPart}{\mathrm{Im}}

\newcommand{\Argument}{\mathrm{Arg}}

\newcommand{\norm}[1]{\| #1 \|}

\newcommand{\inner}[2]{\langle #1, #2 \rangle}

\newcommand{\Span}{\mathrm{span}} \newcommand{\AA}{\unicode[.8,0]{x212B}}

\newcommand{\vectorA}[1]{\vec{#1}}      % arrow

\newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow

\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } 

\newcommand{\vectorC}[1]{\textbf{#1}} 

\newcommand{\vectorD}[1]{\overrightarrow{#1}} 

\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} 

\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}

\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } 

\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}

\newcommand{\avec}{\mathbf a} \newcommand{\bvec}{\mathbf b} \newcommand{\cvec}{\mathbf c} \newcommand{\dvec}{\mathbf d} \newcommand{\dtil}{\widetilde{\mathbf d}} \newcommand{\evec}{\mathbf e} \newcommand{\fvec}{\mathbf f} \newcommand{\nvec}{\mathbf n} \newcommand{\pvec}{\mathbf p} \newcommand{\qvec}{\mathbf q} \newcommand{\svec}{\mathbf s} \newcommand{\tvec}{\mathbf t} \newcommand{\uvec}{\mathbf u} \newcommand{\vvec}{\mathbf v} \newcommand{\wvec}{\mathbf w} \newcommand{\xvec}{\mathbf x} \newcommand{\yvec}{\mathbf y} \newcommand{\zvec}{\mathbf z} \newcommand{\rvec}{\mathbf r} \newcommand{\mvec}{\mathbf m} \newcommand{\zerovec}{\mathbf 0} \newcommand{\onevec}{\mathbf 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}

This work is licensed under a Creative Commons Attribution 3.0 Unported License. You’re free to print it, copy it, translate it, rewrite it, set it to music, slice it, dice it, or whatever, so long as you attribute the original to me, Alex Reinhart, and provide a link back to this site. (If you do translate it, please let me know! I’d happily provide a link to your translation.) Hit the link to the license for more details.

The xkcd cartoon used inside is available under the Creative Commons Attribution-NonCommercial 2.5 License, and may not be used commercially without permission from the author. More details.

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