Space of matrices B for which there is a solution to Bx=c for a given c

Question

Let $F$ be a field, $k$ and $m$ natural numbers with $k \leq m$, and $c \in F^m$.

Is there some name for the set $\mathcal{B}_c = \{ B \in F^{m \times k}\, | \,\, \exists x \in F^k $s.t. $ Bx = c\}$ of all matrices whose columns can be linearly combined to form $c$?

Furthermore, is there some known structure for this set?

I am particulary interested in the intersection of several of such sets (for different $c$'s), in the case where $F$ is finite.

For example, if $F = \mathbb{R}$, $m=3$ and $k=2$, then $\mathcal{B}_c$ contains the set of all matrices whose two columns are vectors $x,y \in \mathbb{R}^3$ such that $c \in span(\{ x,y\})$. This set is sometimes referred to as the plane sheaf for vector $c$. Is there some generalization of this concept for other values of $k$ and $m$ and other fields?

Thanks,

Answer 1

If you are willing to replace the vectors $c$, $x$, and $y$ by the spaces $U_1 = \text{span}(c)$ and $U_2 = \text{span}(x,y)$, which in some sense doesn't change the problem, then one generalization is the concept of a partial flag variety. An example of such is the space of all $U_1\subset U_2\subset F^m$ where the $U_i$ are subspaces of dimension $i$. Your setup corresponds to a subset of this flag variety with $U_1 = \text{span}(c)$.

If you intersect over multiple such $c$ then you just get a smaller subset. Or viewed in another way, you can replace $U_1$ with the span of all the $c$ of interest to get another set of the same form, with $U_1$ now having potentially higher dimension.