Exercise 8, § 3

Let $g$ be the mapping such that, for each matrix $X \in L_{n,p}$, $X'g(X)$ is the matrix formed by the first $q$ rows of $X$ ($q < p$). Show that the restriction of $g$ to $V_{n,p}$ is a continuous open mapping of $V_{n,p}$ onto $V_{n,q}$ [use Exercise 7 a), by noting that if $X = U . Y$ then

where $U'$ is the matrix obtained from $U$ by removing the rows and columns with indices $> q$; on the other hand, observe that $f$ is open].