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Ask a fundamentla problem:

Suppose a matrix $A \in R^{n \times m}$ can be factored into $A = UV'$, with $U \in R^{n \times k}$ and $V \in R^{m \times k}$
If $m,n \geq k$, what the rank of matrix $A$ at most could be? and how to prove it?  

If deleting the condition $m,n \geq k$, then what the rank of matrix $A$ at most could be?

Thanks,

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1 Answer 1

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The rank of $AB$ cannot be more than $A$, as the vectors of the form $A(Bv)$ are a subset (subspace, actually) of those that are of the form $Av$. It also cannot exceed the rank of $B$, as $A$ can at best be one-to-one, and so at best could preserve the dimension of $B(v)$.

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