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"If matrices B and AB have the same rank, prove that they must have the same null spaces."

I have absolutely NO idea how to prove this one, been stuck for hours now. Even if you don't know the answer, any help is greatly appreciated.

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    This has got to be false. Do you mean the _dimensions_ of the null spaces are equal? Wait a minute - maybe it's true. It is easy to show the null space of $B$ is a subset of the null space of $AB$. I see someone just gave an answer, so I'll stop here.2012-10-10

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