Skip to main content\(\newcommand{\nul}{\operatorname{nul}}
\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
\newcommand{\C}{\mathbb C}
\newcommand{\R}{\mathbb R}
\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{\complex}{\mathbb C}
\newcommand{\field}{\mathbb F}
\newcommand{\pbb}{\mathbb P}
\newcommand{\poly}{\mathbb P}
\newcommand{\basis}[2]{#1_1,#1_2,\ldots,#1_{#2}}
\newcommand{\conjugate}[1]{\overline{#1}}
\newcommand{\conj}[1]{\overline{#1}}
\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]{\operatorname{Span}\{#1\}}
\newcommand{\bcal}{{\cal B}}
\newcommand{\ccal}{{\cal C}}
\newcommand{\dcal}{{\cal D}}
\newcommand{\ecal}{{\cal E}}
\newcommand{\scal}{{\cal S}}
\newcommand{\wcal}{{\cal W}}
\newcommand{\ecal}{{\cal E}}
\newcommand{\fcal}{{\cal F}}
\newcommand{\kcal}{{\cal K}}
\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}
\newcommand{\gray}[1]{\color{gray}{#1}}
\newcommand{\lgray}[1]{\color{lightgray}{#1}}
\newcommand{\go}{\gray{1}}
\newcommand{\gz}{\gray{0}}
\newcommand{\rank}{\operatorname{rank}}
\newcommand{\row}{\operatorname{Row}}
\newcommand{\col}{\operatorname{Col}}
\renewcommand{\deg}{\operatorname{deg}}
\newcommand{\range}{\operatorname{range}}
\newcommand{\tr}{\operatorname{tr}}
\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{\inner}[2]{\langle #1,#2\rangle}
\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\,}$}}}
\)
Understanding More Linear Algebra
David Austin
Department of Mathematics
Grand Valley State University
July 20, 2025
