$\newcommand{\br}{\\}$
$\newcommand{\R}{\mathbb{R}}$
$\newcommand{\Q}{\mathbb{Q}}$
$\newcommand{\Z}{\mathbb{Z}}$
$\newcommand{\N}{\mathbb{N}}$
$\newcommand{\C}{\mathbb{C}}$
$\newcommand{\P}{\mathbb{P}}$
$\newcommand{\F}{\mathbb{F}}$
$\newcommand{\L}{\mathcal{L}}$
$\newcommand{\spa}[1]{\text{span}(#1)}$
$\newcommand{\set}[1]{\{#1\}}$
$\newcommand{\emptyset}{\varnothing}$
$\newcommand{\otherwise}{\text{ otherwise }}$
$\newcommand{\if}{\text{ if }}$
$\newcommand{\union}{\cup}$
$\newcommand{\intercept}{\cap}$
$\newcommand{\abs}[1]{| #1 |}$
$\newcommand{\pare}[1]{\left\(#1\right\)}$
$\newcommand{\t}[1]{\text{ #1 }}$
$\newcommand{\head}{\text H}$
$\newcommand{\tail}{\text T}$
$\newcommand{\inv}[1]{{#1}^{-1}}$
$\newcommand{\nullity}[1]{\text{nullity}(#1)}$
$\newcommand{\rank}[1]{\text{rank }#1}$
$\newcommand{\oto}{\text{ one-to-one }}$
$\newcommand{\ot}{\text{ onto }}$
$\newcommand{\Vcw}[2]{\begin{bmatrix} #1 \br #2 \end{bmatrix}}$
$\newcommand{\Vce}[3]{\begin{bmatrix} #1 \br #2 \br #3 \end{bmatrix}}$
$\newcommand{\Vcr}[4]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \end{bmatrix}}$
$\newcommand{\Vct}[5]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \end{bmatrix}}$
$\newcommand{\Vcy}[6]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \br #6 \end{bmatrix}}$
$\newcommand{\Vcu}[7]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \br #6 \br #7 \end{bmatrix}}$
$\newcommand{\Mqw}[2]{\begin{bmatrix} #1 & #2 \end{bmatrix}}$
$\newcommand{\Mqe}[3]{\begin{bmatrix} #1 & #2 & #3 \end{bmatrix}}$
$\newcommand{\Mqr}[4]{\begin{bmatrix} #1 & #2 & #3 & #4 \end{bmatrix}}$
$\newcommand{\Mqt}[5]{\begin{bmatrix} #1 & #2 & #3 & #4 & #5 \end{bmatrix}}$
$\newcommand{\Mwq}[2]{\begin{bmatrix} #1 \br #2 \end{bmatrix}}$
$\newcommand{\Meq}[3]{\begin{bmatrix} #1 \br #2 \br #3 \end{bmatrix}}$
$\newcommand{\Mrq}[4]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \end{bmatrix}}$
$\newcommand{\Mtq}[5]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \end{bmatrix}}$
$\newcommand{\Mqw}[2]{\begin{bmatrix} #1 & #2 \end{bmatrix}}$
$\newcommand{\Mwq}[2]{\begin{bmatrix} #1 \br #2 \end{bmatrix}}$
$\newcommand{\Mww}[4]{\begin{bmatrix} #1 & #2 \br #3 & #4 \end{bmatrix}}$
$\newcommand{\Mqe}[3]{\begin{bmatrix} #1 & #2 & #3 \end{bmatrix}}$
$\newcommand{\Meq}[3]{\begin{bmatrix} #1 \br #2 \br #3 \end{bmatrix}}$
$\newcommand{\Mwe}[6]{\begin{bmatrix} #1 & #2 & #3\br #4 & #5 & #6 \end{bmatrix}}$
$\newcommand{\Mew}[6]{\begin{bmatrix} #1 & #2 \br #3 & #4 \br #5 & #6 \end{bmatrix}}$
$\newcommand{\Mee}[9]{\begin{bmatrix} #1 & #2 & #3 \br #4 & #5 & #6 \br #7 & #8 & #9 \end{bmatrix}}$