Solution
= Solution
\b[The system is consistent for every $b\in\mathbb R^m$:] take $x=Yb$, since $A(Yb)=(AY)b=b$. Thus $Y$ is a <right inverse> giving an explicit solution for every right-hand side.
All solutions are $Yb+u$ with $u$ in the <kernel of a linear map> represented by $A$. The <rank-nullity theorem> gives kernel dimension $n-m$; the solution is unique when $n=m$, and is nonunique when $n>m$.