While reading an article about rank of matrices the following question came to my mind:
If $A$ and $B$ are $n\times m$ matrices and $m\times k$ matrices such that $AB=0$ then what do we know about $rk(A)+rk(B)$?
My guess was that $rk(A)+rk(B)\leq m$, I've tried many examples and all of them works, but I haven't found a proper proof. I'm pretty sure that we have to use that the rank coincides with the number of linearly independent rows and with the number of linearly independent columns. Probably the solution will be short and easy, but I'm missing something.
Thanks for helping.