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$A$ is an $M\times N$ matrix with linearly independent rows and linearly independent columns. Prove that $A$ must be square matrix.

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    Assume wlog that $M. Then $A:\Bbb R^N\to \Bbb R^M$ must have nontrivial kernel...2012-10-10
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    You also can see this by the dimensionforumula for functions in $Hom(\mathbb R^n, \mathbb R^m)$. Use this to conclude that $\dim \ker A >0$ with $im(A) = \mathbb R^m$.2012-10-10
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    $\rank(A)=$ the number of linearly independent columns, and $\rank(A)=$the number of linearly independent rows.......2012-10-10

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