Prove that $\operatorname{rank}A = \operatorname{rank}BA + \dim(\operatorname{Im}A \cap \ker B)$ for any linear operators on $V$

Question

I have the following task (Kostrikin, Linear Algebra, 2.2.5):

Prove that for any linear operators on $V$ the following equation holds: $$\operatorname{rank}A = \operatorname{rank}BA + \dim(\operatorname{Im}A \cap \ker B)$$

UPD. Originally, equation in my question was different, because of the typo in the textbook. After some googling I've found a right one.

Answer 1

Let $B'=B\lvert_{\operatorname{Im}A}$ and apply the rank-nullity theorem:

$\dim \operatorname{Im} A = \dim \operatorname{Im} B'+ \dim \operatorname{Ker}B'$ but it's straightforward to prove $\operatorname{Im} B' = \operatorname{Im} BA$ and $\operatorname{Ker}B' = \operatorname{Ker}B \cap \operatorname{Im}A$.