Let $A$ be an $m \times n$ matrix of rank $r$, let $I$ be a set of row indices such that the corresponding rows of $A$ are independent and let $J$ be a set of $r$ column indices such that the corresponding columns of $A$ are independent. Let $M$ denote the $r \times r$ submatrix obtained by taking rows from $I$ and columns from $J$. Then $M$ is invertible.
So far I've got: Let $B$ be the $m \times r$ matrix obtain from taking the rows from $J$. $B$ is row-reducible to having a pivot in each column or else $Ax = 0$ has more than one solution, a contradiction. Also, the dimension of the row space of $B$ equals $r$ as well.