= Image criterion for a matrix factorization
{title2=$AX=B\iff\operatorname{im}B\subseteq\operatorname{im}A$}
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>.
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