Finding the span of $AB=0 $ if I have the basis of $Ax=0$ I have the basis of a kernel of a matrix $A\in M_{4\times 4}(\mathbb R)$, or the basis of $Ax=0$, suppose the given basis of the kernel is $v_1,v_2$. How can I find the span of $W$ such that $W= \{ B\in M_{4\times 2}(\mathbb R)|AB=0\} $ ?
(I have an example with numbers for the given basis and the matrix $A$ but I would prefer a general solution if it's not a problem)

Fix the matrix $A$, noting $\ker(A) = \text{span}(v_1,v_2)$.  Let $B$ be a matrix such that $AB = 0$.
It follows that for all $x$, $A(Bx) = (AB)x = 0$.  So, $Bx \in \ker(A)$ for every $x$, so the image of $B$ is a subset of the kernel of $A$ if $B \in W$.
Now, suppose $B$ is an arbitrary matrix such that the image of $B$ is in the kernel of $A$.  It follows that for all $x$, we have $(AB)x = A(Bx) = 0$.  Since $(AB)x = 0$ for all $x$, $AB$ must be the zero matrix, so that $B \in W$ if its image is in the kernel of $A$.
So, $W$ is exactly the set of matrices whose image is contained in the kernel of $A$.  That is, $W$ is the set of matrices whose columns are of the form $\alpha v_1 + \beta v_2$ for $\alpha,\beta \in \mathbb{R}$.
