I am considering a basic but general problem: let $A\in \mathbb{R}^{m \times n}, B\in \mathbb{R}^{m \times l}$, where $m, n$ and $l$ are positive integers. What is the condition that there exists an $X \in \mathbb{R}^{n \times l}$ s.t. $B = AX$?
I think a necessary condition is something about the ranks of $A, B$, and $X$: the rank of $B$ has to be equal or less than that of $A$, as $r_B \leq \min (r_A, r_X) \leq r_A$ (where $r_M$ denotes the rank of a matrix $M$).
Or, instead of answering the above, can we consider a more abstract question? By an abstract question, I mean something like the existence of a linear transformations that sends a given $v \in V$ to a given $w\in W$, where $V$ and $W$ are finite dimensional linear spaces.
The matricial equation $AX=B$ can be written as a linear system of $ml$ equations with $nl$ unknowns. You can apply Rouché-Frobenius to discuss it.
Perhaps is more efficient to select $k=\min\{m,n,l\}$ rows of $A$ and $k$ columns of $B$. If the rank of $A$ is $\ge k$, compute $X=B_kA_k^{-1}$ and try to complete it.