Find a basis in $\Bbb{C}^{(2\times 2)}$ over $\Bbb{C}$
Solution 1:
Hint: Try to prove that for any field (even a unital ring) $K$, the following form a basis to $M_2(K)$: \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}
Another way: try to prove that $M_2(K)\cong K^4$ as vector spaces, then look at the image of the basis elements of $K^4$ in $M_2(K)$