If $\operatorname{ker} T \neq \{0\}$, there is $S: V \rightarrow V$, $S\neq 0$, with $T \circ S = 0$.

Question

Let $V$ be a vector space and $T$ be linear transformation $T: V \rightarrow V$ . If $\operatorname{ker} T \neq \{0\}$, prove that there's a linear transformation $S: V \rightarrow V$ such that $S$ is not the zero transformation but $T \circ S = 0$.

is there any sentence or a any direction that anyone can give me ? I have no clue what to do(I could prove that The second side will prove the $\operatorname{ker} T$ is not zero but i dont have any idea to prove the first side...}

Answer 1

Hint 1: $T \circ S = 0$ if and only if for every $x \in V$, $S(x) \in \ker T$.

Hint 2: For any vector spaces $U,V$ for which $U \neq \{0\}$ and $V \neq \{0\}$, there exists a linear transformation $S: U \to V$ with $S \neq 0$.