Prove: $T \in L(V, V)$ then $ \exists S \in L(V,V)$ such that $ST = 0 \iff T$ is not onto
Proof: $\rightarrow$
Let $S \in L(V,V)$ s.t $S \neq 0$ and $ST = 0$. Consider $S(T(v))$ for some $v\in V$ Then $T=0$ and we have $S(T(v)) = S(0) = 0. \iff$ is not one to one$\iff T$ is not onto
- Is this correct so far?
- I need help with the other direction
- I think I can just take the reverse steps if this is correct