Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question

1 Answer 1

up vote 1 down vote accepted

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.

share|improve this answer
    
Thanks, I got it! –  Alberto Oct 30 '12 at 0:48

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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