Let $\sigma_0,\sigma_x,\sigma_y,\sigma_z$ stand for the $2\times 2$ identity matrix and the well known pauli matrices:
\begin{equation} \sigma_0=\left(\begin{array}{cc}1&0\\0&1\end{array}\right), \sigma_x=\left(\begin{array}{cc}0&1\\1&0\end{array}\right), \sigma_y=\left(\begin{array}{cc}0&-i\\i&0\end{array}\right), \sigma_z=\left(\begin{array}{cc}1&0\\0&-1\end{array}\right) \end{equation}
How would you prove that the $4\times 4$ Hermitian matrices constitute a linear vector space with basis the tensor products of $\sigma_0,\sigma_x,\sigma_y,\sigma_z$?