\(\newcommand{\ds}{\displaystyle} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\ba}{\left[\begin{array}} \newcommand{\ea}{\end{array}\right]} \newcommand{\normal}{\trianglelefteq} \newcommand{\Span}{\operatorname{Span}} \newcommand{\Null}{\operatorname{Null}} \renewcommand{\Im}{\operatorname{Im}} \newcommand{\nullity}{\operatorname{nullity}} \newcommand{\rank}{\operatorname{rank}} \newcommand{\range}{\operatorname{range}} \newcommand{\tr}{\operatorname{tr}} \newcommand{\proj}{\operatorname{proj}} \newcommand{\col}{\operatorname{Col}} \newcommand{\row}{\operatorname{Row}} \newcommand{\diag}{\operatorname{diag}} \newcommand{\rref}{\operatorname{rref}} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\sgn}{\operatorname{sgn}} \newcommand{\0}{\mathbf 0} \newcommand{\E}{\mathbf E} \newcommand{\e}{\mathbf e} \newcommand{\bdot}{\boldsymbol \cdot} \newcommand{\C}{\mathbb C} \newcommand{\Q}{\mathbb Q} \newcommand{\R}{\mathbb R} \newcommand{\Z}{\mathbb Z} \newcommand{\cB}{\mathcal B} \newcommand{\cC}{\mathcal C} \newcommand{\cE}{\mathcal E} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)