The dimension formula for the largest matchable subspace

Let $K\subset F$ be a field extension, and let $A$ and $B$ be two $n$-dimensional $K$-subspaces of $F$, where $n\geq 1$. Write $\langle AB\rangle$ for the $K$-linear span of the products of elements of $A$ and $B$, and let $M(A,B)$ denote the dimension of the largest subspace of $A$ that is matched to a subspace of $B$. For a basis $\mathcal{A}=\{a_1,\ldots,a_n\}$ of $A$, define

$$D_{\mathcal{A}}(B)=\max\left\{\#J:J\subset\{1,\ldots,n\},\ \dim\bigcap_{i\in J}\left(a_i^{-1}A\cap B\right)>n-\#J\right\},$$

and define

$$D(A,B)=\max\{D_{\mathcal{A}}(B):\mathcal{A}\text{ is a basis for }A\}.$$

The dimension formula for the largest matchable subspace. If $\langle AB\rangle\neq A$ and $A$ is not matched to $B$, then

$$M(A,B)=n-D(A,B).$$

This is the proposed linear analogue of the corresponding result for unmatchable subsets in groups. It relates the failure of the matching condition to the dimension of the largest subspace of $A$ that can still be matched to a subspace of $B$; the supplied text does not establish the formula or provide evidence resolving it.