$A\in M\_{m,n}(F)$ we know that $\dim\ker A +\dim \operatorname{ran} A=n$. Suppose $\dim \ker A=r$,and $\{v\_{1},...,v\_{r}\}$ is a basis for the nullspace of A.It can be extended to a basis
$\{v\_{1},...,v\_{n}\}$ of $F^{n}$.
how to prove that $Av\_{r+1},...,Av\_{n}$ is a basis for the range of A?