On the null spaces of projection operators having the same range

Question

Let $K$ be a field of characteristic zero.

If $T,S: K^n \to K^n$ be linear operators such that $T^2=T, S^2=S$ and range$(T)=$range$(S)$, then is it true that $\ker T=\ker S$ ?

Since $T,S$ have same minimal polynomial $x(x-1)$, are diagonalizable and have the same rank , so $T$ and $S$ are conjugate to each other. But I'm not sure what I can say about their null space ...

Answer 1

Let $n=2$ and $\{u,v\}$ be a basis of $K^2$. Define $Su=Tu=u$ and $Sv=T(u+v)=0$. Then $S^2=S,\,T^2=T$ and $\operatorname{range}(S)=\operatorname{range}(T)=\operatorname{span}(u)$, but $\ker(S)=\operatorname{span}(v)\ne\operatorname{span}(u+v)=\ker(T)$.