$Ax=0$ has a nontrivial solution iff $\rank(A)<m$

Let $A\in M_{m\times n}(F)$. Prove that the homogeneous system \[ Ax=0 \] has a nonzero solution if and only if \[ \operatorname{rank}(A)<n. \]

Solution

By rank-nullity for the linear map $A:F^n\to F^m$, \[ n=\dim\ker A+\operatorname{rank}(A). \] Hence \[ \ker A\ne0 \iff \dim\ker A>0 \iff \operatorname{rank}(A)<n. \] A nonzero vector in $\ker A$ is exactly a nontrivial solution of $Ax=0$.