Let $V=M_{22}$ and let $L$:$VV$ be the linear op$-$ erator defined by $L(A)=A^T,$ for $A$ in $V.$ Let $S=\{A_1,A_2,A_3,A_4\},$ where $A_1=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]0,A_2=\left[\begin{array}{ccc}0& 1& 0\end{array}\right]0,A_3=\left[\begin{array}{ccc}0& 0& 0\end{array}\right]1,$ and $A_4=\left[\begin{array}{ccc}0& 0& 1\end{array}\right]0,$ be a basis for $V.($a$)$ Find $[L(A_i){]}_S$ for $i=1,2,3,4.($b$)$ Find the matrix $B$ representing $L$ with respect to $S.($c$)$ Find the eigenvalues and associated eigenvectors of B$.$