# Function from one Null space to Another

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 $A,B$.

Presumably, we're looking at the mappings from $R$ to $R$ that happen to take the null space of $A$ to the nullspace of $B$. Presumably, $A,B:R \to S$, where $S$ is some space we don't care about.
We can build such a mapping as follows: suppose $R$ is $n$ dimensional, and that the nullspaces of $A$ and $B$ have dimensions $a$ and $b$ respectively. Take $\{v_1,\dots,v_a\}$ and $\{w_1,\dots,w_b\}$ to be some (orthonormal) bases for the nullspaces of $A$ and $B$ respectively. Extend these so that $\{v_1,\dots,v_n\}$ and $\{w_1,\dots,w_n\}$ are (orthonormal) bases of $R$.
Define the following matrices: $$V_a = (v_1 \cdots v_a)\\ V_a' = (v_{a+1} \cdots v_n)\\ V = (V_a\; V_a')\\ W_b = (w_1 \cdots w_b)\\ W_b' = (w_{b+1} \cdots w_n)\\ W = (W_b\; W_b')$$ Define $$M = \pmatrix{T_1&0\\ T_2&T_3}$$ where $T_1$ is $b \times a$ and $M$ is $n \times n$
Then any satisfactory mapping can be expressed as $WMV^{-1}$ for some $M,W,V$ as above. Note that if $\{v_1,\dots,v_n\}$ is orthonormal, then $V^{-1} = V^*$ (or $V^T$ if entries are real).