Let $A,B:V\rightarrow V$. If $ \operatorname{Im}(A)\subseteq \operatorname{Ker}(B)$ then $(AB)^{2}=0$
Intuitively I can see why this holds, but I cannot find the way to prove it.
2026-04-01 17:27:18.1775064438
$ \operatorname{Im}(A)\subseteq \operatorname{Ker}(B)\Rightarrow (AB)^{2}=0$
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Hint: $$(AB)^2=(AB)(AB)=A(BA)B.$$ Can you compute $BA$ from the hypothesis?