4
$\begingroup$

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

$\endgroup$
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.