if T is normal - Im(T) orthogonal to Ker(T)

Question

Hi I want to prove / contradict that if T is normal then Im(T) orthogonal to Ker(T).
Above C it is clear, because each of the Eigenspace are orthogonal to each other and there exist a base of eigenvectors for the whole space.

can I somehow make this insight about all fields and not only C?

Answer 1

If $T$ is normal, then $(Tu,Tu)=(T^*Tu,u)=(TT^*u,u)=(T^*u,T^*u)$. Hence $\operatorname{Ker}T=\operatorname{Ker}T^*$, and if $u=Tw\in\operatorname{Im}T$ and $v\in\operatorname{Ker}T$, then $(u,v)=(Tw,v)=(w,T^*v)=0$.