If $A$ is a real $n\times r$-matrix such that its column vectors are linearly independent, and $B$ be a real $r\times n$-matrix such that its row vectors are linearly independent. Is any of the two following statements true?
a) $\mathrm{ran}(AB) = \ker(B)$.
b) $\mathrm{ran}(AB) = \mathrm{ran}(A)$.
I was thinking that we know that $\mathrm{ran}(A)=0$ for the first one, but I do not see how one can proceed, I feel that I do not know enough about $A$ or $B$ to justify or falsify the statements. I guess for the second one the first step is to prove that $\mathrm{ran}(B)=0$, but I do not see why this is true either.
This is homework, so any tips or suggestions would be good. Or some references to where I can read more about this.