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,