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Question:

A submatrix $B$ of rank $q$ consisting of $s$ rows of $A$ is selected from an $m$-square matrix $A$ of rank $r$. Prove that $q$ is equal to or greater than $r+s−m$.

My thoughts:

I should start by getting $0 \leq q \leq r \leq m$ and $0 \leq q \leq s \leq m$. Can anyone give me guidance on how to proceed?

Note: There was a similar question that has been asked here, but the hint did not help me.

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Note that this is equivalent to $$m-r \geq s-q, $$ the difference between rows and rank of $A$ is greater than in $B$. Now, suppose $A: k^n\to k^m$ where $k$ is your favourite field. Let $U\subseteq k^m$ be the subspace given by the first $s$ coordinates. Then $im(B)=U\cap im(A)$ under the natural embedding.

Let $V=im(A)+U$, then from the second isomorphism theorem we have $$V/im(A) \cong U/im(B) $$ and upon taking dimensions we get $$dim(V) -r = s-q. $$ The inequality follows.

Edit: Here is a simpler solution,

rearrange to get $$r-q\leq m-s. $$ So the difference in rank is less than the difference in rows. There are $r-q$ rows that add the the rank of $A$ but not to the rank of $B$. These rows cannot be in the $s$ selected rows since then this would increase the rank of $B$. Therefore, they are one of the $m-s$ rows not in $B$. The inequality follows.

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