$S$ and $T$ are idempotent linear operators in $V$ which is a vector space in $\mathbb{C}$. Prove that if $S+T$ is idempotent then $ST=TS=0$. If $S+T$ is idempotent, then
$$(S+T)^2=S+T$$
$$\implies S^2+ST+TS+T^2=S+T$$
$$\implies S+T+ST+TS=S+T$$
$$\implies ST+TS=0$$
Now, from here how do I show that $ST=TS=0$?

Essentially equivalent answer but more elegant solution: $ST = SST = -STS = TSS = TS$, so $2ST = ST+TS = 0$ and hence $ST = 0$. Note that you do not need the field $F$ underlying the vector space to be $ℂ$, but you do need $\text{char}(F) ≠ 2$ otherwise you cannot divide by $2$.
