Answer
See below
Work Step by Step
a) Obtain $T(\begin{bmatrix}
1 & 0\\
0 & 0
\end{bmatrix})=(1,1)\\T(\begin{bmatrix}
0 & 1\\
0 & 0
\end{bmatrix})=(0,0)\\T(\begin{bmatrix}
0 & 0\\
1 & 0
\end{bmatrix})=(0,0)\\T(\begin{bmatrix}
0 & 0\\
0 & 1
\end{bmatrix})=(1,1)$
then we have:
$T(E_{11})=(1,1)=1.(1,0)+1.(0,1)\\
T(E_{12})=(0,0)=0.(1,0)+0.(0,1)\\
T(E_{21})=(0,0)=0.(1,0)+0.(0,1)\\
T(E_{22})=(1,1)=1.(1,0)+1.(0,1)$
as $[T(E_{11}]_C=\begin{bmatrix}
1 \\
1
\end{bmatrix}\\
[T(E_{12}]_C=\begin{bmatrix}
0 \\
0
\end{bmatrix}\\
[T(E_{21}]_C=\begin{bmatrix}
0\\
0
\end{bmatrix}\\
[T(E_{22}]_C=\begin{bmatrix}
1 \\
1
\end{bmatrix}$
Hence, $[T]_B^C=\begin{bmatrix}
1 & 0 & 0 & 1\\
1 & 0 & 0 & 1
\end{bmatrix}$
b) Obtain $T(\begin{bmatrix}
-1 & -2\\
-2 & -3
\end{bmatrix})=(-4,-4)\\T(\begin{bmatrix}
1 & 1\\
2 & 2
\end{bmatrix})=(3,3)\\T(\begin{bmatrix}
0 & -3\\
2 & -2
\end{bmatrix})=(-2,-2)\\T(\begin{bmatrix}
0 & 4\\
1 & 0
\end{bmatrix})=(0,0)$
then we have:
$T(E_{11})=(-4,-4)=-4.(1,0)-4.(0,1)\\
T(E_{12})=(3,3)=3.(1,0)+3.(0,1)\\
T(E_{21})=(-2,-2)=-2.(1,0)-2.(0,1)\\
T(E_{22})=(0,0)=0.(1,0)+0.(0,1)$
as $[T(E_{11}]_C=\begin{bmatrix}
-4 \\
-4
\end{bmatrix}\\
[T(E_{12}]_C=\begin{bmatrix}
3\\
3
\end{bmatrix}\\
[T(E_{21}]_C=\begin{bmatrix}
-2\\
-2
\end{bmatrix}\\
[T(E_{22}]_C=\begin{bmatrix}
0\\
0
\end{bmatrix}$
Hence, $[T]_B^C=\begin{bmatrix}
-4 & 3 & -2 & 0\\
-4 & 3 & -2 & 0
\end{bmatrix}$
