Chapter 3.2, Problem 18E

In ${ℝ}^{2\times 2}$ , let

$E_{11}=\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right),E_{12}=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$

$E_{21}=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right),E_{22}=\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right)$

Show that $E_{11},E_{12},E_{21},E_{22}$ span ${ℝ}^{2\times 2}$ .