Chapter 5.4, Problem 30E

With $V$ and $Q$ as in Exercise $29$, find ${\left[A\right]}_Q$ for,

$A=\left[\begin{array}{cc}7& 3\\ -3& -1\end{array}\right]$.

In the vector space $V$ of $\left(2\times 2\right)$ matrices, let $Q=\left\{A_1,A_2,A_3,A_4\right\}$ where

$A_1=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$, $A_2=\left[\begin{array}{cc}1& -1\\ 0& 0\end{array}\right]$,

$A_3=\left[\begin{array}{cc}0& 2\\ 0& 0\end{array}\right]$, and $A_4=\left[\begin{array}{cc}-3& 0\\ 2& 1\end{array}\right]$.

Use the corollary to theorem $5$ and the natural basis for $V$ to show that $Q$ is a basis for $V$.