Image criterion for a matrix factorization ($AX=B⟺\mathrm{im}B\subseteq \mathrm{im}A$)

For matrices representing linear maps on finite-dimensional vector spaces, $AX=B$ has a solution precisely when the image of a linear map represented by $B$ is contained in that represented by $A$. Necessity follows by applying the equation to vectors. For sufficiency choose an $A$-preimage of each column of $B$ and assemble those vectors as the columns of $X$. All solutions are $X_0+N$ with $AN=0$, equivalently every column of $N$ lies in the kernel of a linear map represented by $A$. When $B$ has at least one column, a solution is unique precisely when $A$ has trivial kernel.