If k is an integer between 0 and min(m,n),show that the set of(adsbygoogle = window.adsbygoogle || []).push({});

mxn matrices whose rank is at least k is an open submanifold of

M(mxn, R).Show that this is not true if "at least k"is replaced by

"equal to k."

For this problem, I don't understand why the statement is not true if we replace "at least k" by "equal to k", I am thinking that the set of m*n matrices whose rank is equal to k will have some k*k minor nonsingular, thus any such rank k matrices will have a open nbd contained in Mk(m*n, R) (which is the set of m*n matrices whose rank is equal to k) , and thus make Mk(m*n, R) open in M(m*n,R) (which is the set of all matrices with real entries).

Could anyone help me with the second part of this problem?

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# A question about vector space manifold

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