Prove/Disprove: $S$ not invertible and $\dim(\ker(T)+\dim(\ker(S)\geq\dim(V)$

Question

Let $V$ be a finite vector space and $S,T:V\to V$ linear transformations S.T $T\circ S=0$

Prove/Disprove:

1.if $T\neq 0$ then $S$ is not invertible

2. $\dim(\ker(T)+\dim(\ker(S)\geq\dim(V)$

1. Let assume that $S$ is invertible so there is $S^{-1}$ such that

$$T\circ S\circ S^{-1}=0\iff T=0$$ contradiction

2. seems wrong but I can not find an example

Answer 1

The condition that $TS=0$ implies that $$ \dim(\ker(TS))=\dim(V) $$ Can you convince yourself that $\dim(\ker(TS))\leq \dim(\ker(T))+\dim(\ker(S))$?