If $A, A' \in M_{m*n}(F)$ and $B \in M_{n*p}(F)$ for some field $F$.
Prove:
$Rsp(A) = Rsp(A') \implies Rsp(AB) = Rsp(A'B)$
I want to prove the above, my idea for the proof is to list the matrices in terms of their row vectors and use the Exchange Lemma to show the spans of the row vectors are equal.
However, I am not sure how to correctly lay it out, is my approach correct or is there a different approach I should be taking?
I'd prefer a hint as opposed to a solution at first, but if I can't get it then a solution would be appreciated.