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)

$\mathrm{Im}(B)\subset \mathrm{span}(v1,v2)$ So every column of $B$ must be of the form $av_1+bv_2$.
A Basis of the subspace is therfore given by the matrices $(v_1~0~0~0),(v_2~0~0~0),(0~v_1~0~0),\dots$
