Date: Nov 28, 2012 3:56 PM
Author: Kaba
Subject: Matrices of rank at least k
Hi,

An exercise in a book on smooth manifolds asks me to prove that

(m x n)-matrices (over R) of rank at least k is an open subset of

R^{m x n} (and thus an open submanifold). It is intuitively clear to me

why that is true: an arbitrary small perturbation can add one or more to

the rank of a matrix, but if a matrix is of rank k, then there is a

small open neighborhood in which the rank stays the same. So I should be

able to find a small open neighborhood around each at-least-k rank

matrix which still stays in the set, therefore proving the claim. How do

I find such a neighborhood?

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