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Let $m$ and $n$ be positive integers such that $m<n$ and let $M(m\times n,\mathbb{R})$ be he set of all $m\times n$ matrices with entries from $\mathbb{R}$. Let $M_{m}$ be the subset of $M(m\times n,\mathbb{R})$ consisting of the matrices of rank $m$. Show that $M_{m}$ is an open subset of $M(m\times n,\mathbb{R})$. I know from Linear Algebra that an element of $M_{m}$ must have an invertible $m\times m$ submatrix. I am pretty sure that this fact, along with the fact that the set of invertible $m\times m$ matrices with entries from $\mathbb{R}$, $GL(m,\mathbb{R})$, is an open set, is used in the proof, but I cant connect them together.

Eigenfield
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2 Answers2

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Let $F$ be the function taking $ A \in M(m \times n) $ to the $F(A) = \Sigma_ B \left | det(B) \right | $ where the Sigma runnig over all m by m sub matrices of A. Clearly $F$ is continuous so $F^{ -1} (R \setminus \{ 0 \} )$ is an open set is $ M(m \times n) $

Red shoes
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You are on the right track. Note that an $m\times n$ matrix is of full row rank if and only if some of its $m\times m$ submatrix is invertible. Also, note that any union of open sets is open.

user1551
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