I am following Lee's book on smooth manifolds. On pages 19 and 20 he writes the following:
Suppose $m < n$, and let $D_m \subset M_{m\times n}(\mathbb{R})$ be the set of real $m\times n$ matrices of full rank. Suppose $A \in D_m$. The fact that rank $A = m$ means that $A$ has some nonsingular
$m \times m$ submatrix. By continuity of the determinant function, this same submatrix as nonzero determinant on a neighborhood of $A$ in $D_m$.
I understand everything up to the last sentence. If this is some useful property of continuity or submatrices, I'd like to learn it. Can you explain?
Best Answer
$A$ has some particular submatrix with nonzero determinant. Consider the composition $D_m \to M_{m\times m}(\mathbb{R}) \to \mathbb{R}$ where the first map projects onto that particular submatrix and the second map is the determinant. The preimage of $\mathbb{R}-\{0\}$ is an open neighborhood of $A$ in $D_m$ consisting of all the matrices with that same nonsingular submatrix.