Intuition behind $\ker(T)=\ker(T^*)$ for $T$ a normal operator

Question

Let $T : V \to V$ be a normal operator and $V$ a finite-dimensional vector space. Show that $\ker(T)= \ker(T^*)$ and $\text{im}(T) = \text{im}(T^*)$.

I know how to rigorously show this, but I'm curious if anyone has an intuitive way of understanding why this has to be the case.