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.



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.


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