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.

I have a problem. I wish to understand how following can be proven to hold; $\langle P_S(v), u_m \rangle = \sum_{j=1}^n \frac{ \langle v, u_j \rangle}{||u_j||^2} \langle u_j,u_m \rangle = \langle v , u_m \rangle $, where $P_S(v) =\sum_{j=1}^n \frac{ \langle v, u_j \rangle}{||u_j||^2} u_j $ and Suppose V is a complex inner product space and $B={u_1, u_2,..., u_n }$ is orthogonal set in $V$ with $u_j<>0$ for all j. also prove that $P_S(v)$ is a linear transformation.

share|improve this question
Oh boy, that was pretty unreadable! Please put the math inside dollar signs, and use \langle and \rangle instead of greater/less than signs. –  Hans Lundmark Nov 2 '10 at 12:03
Did you try to compute the left-hand side? –  AD. Nov 2 '10 at 13:43
The tag is misleading, there is a <inner-product-space> tag. I suggest you should change that. –  AD. Nov 2 '10 at 13:46
I am not satisfy this answer please can you elaborate this question as soon as possible. –  user37273 Aug 5 '12 at 13:32

1 Answer 1

The first equality follows by definition. Now notice that $$\langle u_j, u_m \rangle = 0$$ if $i\neq j$.

Thus the sum reduces to $$\frac{\langle v, u_m \rangle}{||u_m||^2}\langle u_m, u_m \rangle $$

Now what is the definition of the norm?

share|improve this answer
Definition of the norm is $ ||u|| = \sqrt{ \langle u,u \rangle }$ –  alvoutila Nov 2 '10 at 14:04
@alvoutila: Good, now you can finish the proof on your own. And you can also take the opportunity to learn what the phrase "rhetorical question" means. ;-) en.wikipedia.org/wiki/Rhetorical_question –  Hans Lundmark Nov 2 '10 at 14:25

Your Answer


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.