Suppose a single vector space over $R$ of degree $n$, and two matrices $A, B$ of arbitrary row size, but col size $n$, s.t. their individual null spaces are linear subspaces of this vector space. Is there a way to linearly represent all linear functions (i.e. matrices) which take the subdomain of $null(A)$ into a subrange of $null(B)$ (does not have to be vice versa, assume that $\left|null(B)\right| \geq \left|null(A)\right|$)?
Let us assume that the effect of such functions is irrelevant on vectors which are not in $null(A)$ at the moment. In addition, my actual difficulty is to do that without using the null-space vector extraction explicitly, but only using $A,B$.