设线性空间 $R^{2 \times 2}$ 上的线性变换 $T$ 为

$$
\begin{gathered}
T(X)=A X, \quad \forall X \in R^{2 \times 2} \\
A=\left(\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right)
\end{gathered}
$$

求 $T$ 在基 $E_{11}=\left(\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right), E_{12}=\left(\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right), E_{21}=\left(\begin{array}{ll}0 & 0 \\ 1 & 0\end{array}\right), E_{22}=\left(\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right)$ 下的矩阵．