Answer
$[E_{12}]_C=(0,0,0,1)\\
[E_{22}]_C=(1,0,0,0) \\
[E_{21}]_C=(0,0,1,0) \\
[E_{11}]_C=(0,1,0,0)\\
P_{C \leftarrow B}=\begin{bmatrix}
0 &1 & 0 & 0\\
0 &0 & 0 & 1\\
0 &0 & 1 & 0\\
1 &0 & 0 & 0
\end{bmatrix}$
Work Step by Step
We are given:
$B=\{E_{12},E_{22},E_{21},E_{11}\}$
$B=\{E_{22},E_{11},E_{21},E_{12}\}$
To find: $[E_{12}]_C=0E_{22}+0E_{11}+0E_{21}+1E_{12} \\
[E_{22}]_C=1E_{22}+0E_{11}+0E_{21}+0E_{12}\\
[E_{21}]_C=0E_{22}+0E_{11}+1E_{21}+0E_{12}\\
[E_{11}]_C=0E_{22}+1E_{11}+0E_{21}+0E_{12}\\
$
Hence, we have the vectors:
$[E_{12}]_C=(0,0,0,1)\\
[E_{22}]_C=(1,0,0,0) \\
[E_{21}]_C=(0,0,1,0) \\
[E_{11}]_C=(0,1,0,0)\\
P_{C \leftarrow B}=\begin{bmatrix}
0 &1 & 0 & 0\\
0 &0 & 0 & 1\\
0 &0 & 1 & 0\\
1 &0 & 0 & 0
\end{bmatrix}$
