Exercise 6, § 10

Let $X$ be a matrix with $m$ rows and $n$ columns over a field $K$; show that the rank $\operatorname{rg}(X)$ is equal to the greatest of the ranks of the submatrices of $X$ with an equal number of rows and columns. (Let $\operatorname{rg}(X) = r$; if $a_1, \ldots, a_r$ are $r$ columns of $X$ forming a free system in $K_d^m$ and a basis of $K_d^m$ is formed with these $r$ vectors and $m - r$ vectors of the canonical basis, show that the components of $a_1, \ldots, a_r$ over the $r$ other vectors of the canonical basis form a matrix of rank $r$.)