$\begin{bmatrix}a&b\\\\c&d\end{bmatrix} \longrightarrow \begin{bmatrix}a\\\\a\\\\3a+b\end{bmatrix}$
As the kernel maps everything to zero then $a=0$ and $3a+b=0$ so the basis is of the form
$\left\{\begin{bmatrix}0&0\\\\1&0\end{bmatrix},\begin{bmatrix}0&0\\\\0&1\end{bmatrix}\right\}$
However i'm not sure if this is correct so far