If Same Rank, Same Null Spaces? "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.

I would begin by showing that the null space of $B$ is a subspace of the null space of $AB$. Next show that having the same rank implies they have the same nullity. Finally, what can you conclude when a subspace is the same dimension as its containing vector space?
