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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 $A,B$.

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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).

I don't think that you can avoid "nullspace extraction" on some level, unfortunately.

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  • $\begingroup$ Thank you! of course, when extracting null space, you can specifically parametrize the entire space of such matrices, and otherwise you might be able to work with projection operators, but nothing so simple, unfortunately. $\endgroup$ – Amir Vaxman Sep 23 '14 at 14:45

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