Let $A\in\Bbb F^{m×n}$ be a matrix with $rank(A)=n$. Prove $A$ is left-invertible: there exists a matrix $B\in\Bbb F^{n×m}$ such that $BA=I_n$.
I know $rank(BA)≤\min(rank(A),rank(B))=\min(n,rank(B))$. The rank of $B$ can't be less than $n$ otherwise $BA$ won't be a full-rank matrix and hence won't be row-equivalent to $I_n$. so $n≤rank(B)$ and therefore $rank(BA)≤n$. How can I make it so $rank(BA)=n$?