TITLE: Direct decomposition of vector space in image of map plus kernel of adjoint
QUESTION [6 upvotes]: 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

REPLY [3 votes]: Use the obvious fact that $\ker A^*=(\mathrm{Im}\ A)^{\perp}$. Now it remains to show that $W=\mathrm{Im}\ A\oplus(\mathrm{Im}\ A)^{\perp}$. But this follows from the definition of the orthogonal complement.