Nullity of linear transformation I'm struggling to find the nullity $N(T)$ of the following linear transformation (in the canonical basis of $\mathbb{R^{2\times2}}$

$
M = \begin{bmatrix}
       0 &amp; 0 &amp; 0 &amp; 0           \\[0.3em]
       -1 &amp; -3 &amp; 3 &amp; 0  \\[0.3em]
       0 &amp; 0 &amp; 1 &amp; 0  \\[0.3em]
       0 &amp; 0 &amp; 0 &amp; 1 
     \end{bmatrix}
$

What's making me feel confused is that we are considering a basis of matrices. I knew how proceed in the case of for example the canonical basis of $\mathbb{R^2}$ (just multiply by a column vector (a,b,c,d) and the I would have a system of 4 linear equations (equal to zero)). But the presence of matrices 2x2 is driving me confused because I can't even try to multiply my matrix of the linear transformation to a matrix 2x2 

Am I making myself clear? If you could help me, please...

Thanks
Hint:

A matrix 
$$
A=
\begin{bmatrix}
a&amp;b\\c&amp;d
\end{bmatrix}
$$
is represented in the canonical basis as:
$$
A=a\begin{bmatrix}
1&amp;0\\0&amp;0
\end{bmatrix}
+b\begin{bmatrix}
0&amp;1\\0&amp;0
\end{bmatrix}
+c\begin{bmatrix}
0&amp;0\\1&amp;0
\end{bmatrix}
+d\begin{bmatrix}
0&amp;0\\0&amp;1
\end{bmatrix}
$$
so,  in this basis, you can represent it as a vector:
$$
\vec A=
\begin{bmatrix}
a\\b\\c\\d
\end{bmatrix}
$$ 
So your matrix $M$ acts as:
$$
M\vec A=
\begin{bmatrix}
       0 &amp; 0 &amp; 0 &amp; 0   \\       
       -1 &amp; -3 &amp; 3 &amp; 0  \\
       0 &amp; 0 &amp; 1 &amp; 0  \\
       0 &amp; 0 &amp; 0 &amp; 1 
     \end{bmatrix}\begin{bmatrix}
a\\b\\c\\d
\end{bmatrix}
$$
and you have to solve :
$$
\begin{bmatrix}
       0 &amp; 0 &amp; 0 &amp; 0   \\       
       -1 &amp; -3 &amp; 3 &amp; 0  \\
       0 &amp; 0 &amp; 1 &amp; 0  \\
       0 &amp; 0 &amp; 0 &amp; 1 
     \end{bmatrix}\begin{bmatrix}
a\\b\\c\\d
\end{bmatrix}=0
$$
If $\mathbf{x}=(x,y,z,w)$, then right away you should see that $z=w=0$ and $x=-3y$. So the nullity has basis: $(-3,1,0,0)$ and is of dimension $1$.

If $\mathbf{x}$ is really a $2\times 2$ matrix, then you can think of it as $\left[\begin{matrix} x &amp; y \\ z &amp; w\end{matrix}\right]$. Converting between a matrix and its representation as a vector should be clear.