Given $A,B \in Hom(V)$, if $AB(v)=0, \forall v \in V$, prove that $BA=0$ ($V$ is a finite vector space)

Question

$A[BA(v)]=AB[A(v)]=0 \implies A[BA(v)]=0 \implies ( BA(v)=0, \forall v \in V ) \vee ( BA(v) \in Ker(A), \forall v \in V )$

I'm unsure how to prove that $\exists v\in V:BA(v)\notin Ker(A)$

Answer 1

That statement is false. If $V=F^3$, for some field $F$, if $A(x,y,z)=(0,z,0)$, and if $B(x,y,z)=(y,0,0)$, then you always have $A\bigl(B(v)\bigr)=0$, but $B\bigl(A(0,0,1)\bigr)\ne(0,0,0)$.