Let $A:V\to W$ be a linear map with $V,W$ finite dimensional Hilbert spaces. Is it always true that $ \dim(\mathrm{Im}(A)) + \dim(\ker(A^*)) = \dim(W),$
i.e. (since $\mathrm{Im}(A) \cap \ker(A^*) = 0$) $W = \mathrm{Im}(A) \oplus \ker (A^*)?$
Notation: $A^*$ is the adjoint of $A$, $\mathrm{Im}$ and $\ker$ stand for Image and Kernel.
I have something like this in mind, but don't find it in my linear algebra notes.
Thanks