I'm thinking about the intersection of the null space of matricies $\mathcal{A}$ and $\mathcal{B}$.
The null space of each matrix is:
$\mathcal{A}x = 0$
$\mathcal{B}y = 0$
Id like to find the intersection of these two null spaces. Hence I want to find the subspace of all vectors $z$, s.t. {$z \in \mathcal{R} | Az = 0\mbox{ and }Bz = 0$}.
Now I already know that if I define a new matrix $C = [A;B]$ (col stacked) then the $z = \{Cz=0|z\in\mathcal{R}\}$ is in the null space intersection.
However,
$\mathcal{A}z = 0$
$\mathcal{B}z = 0$
Hence, $\mathcal{A}z = \mathcal{B}z$, and hence $(\mathcal{A}-\mathcal{B})z = 0$. Therefore the intersection of the two null spaces is really the null space of the subtraction of the two matricies.
Is this true?