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.

explain $\langle \sum_{j=1}^n \langle v,u_j \rangle u_j, \sum_{k=1}^n \langle w,u_k \rangle u_k \rangle =\sum_{j=1}^n \langle v,u_j \rangle \sum_{k=1}^n \langle u_j, \langle w, u_k \rangle u_k \rangle$, V is complex inner product space with orthonormal basis $R={u_1, u_2,..., u_n}$

share|improve this question
    
Explain what? How you get from the left hand side to the right hand side? –  Arturo Magidin Nov 2 '10 at 14:28
    
As in your question math.stackexchange.com/questions/8632/… I suggest you enter the proper tag. –  AD. Nov 2 '10 at 20:05
    
-1: This is a bit of a one-way communication is it not? Also, you don't read our comments or accept the answers, do you? –  AD. Nov 2 '10 at 20:14

1 Answer 1

up vote 3 down vote accepted

The inner product is sesquilinear, and each $\langle v,u_j\rangle$ and $\langle w,u_k\rangle$ are scalars. Rewrite $\alpha_j = \langle v,u_j\rangle$ and $\beta_k = \langle w,u_k\rangle$. Then the left hand side is $$\left\langle \sum_{j=1}^n\langle v,u_j\rangle u_j, \sum_{k=1}^{n}\langle w,u_k\rangle u_k\right\rangle = \left\langle \sum_{j=1}^n \alpha_ju_j,\sum_{k=1}^n\beta_k u_k\right\rangle.$$ So by additivity and homogeneity in the first component you have: \begin{eqnarray*} \left\langle \sum_{j=1}^n\langle v,u_j\rangle u_j, \sum_{k=1}^n\langle w,u_k\rangle u_k\right\rangle & = & \left\langle \sum_{j=1}^n \alpha_ju_j,\sum_{k=1}^n\beta_k u_k\right\rangle\\ & = & \sum_{j=1}^{n}\left\langle \alpha_j u_j, \sum_{k=1}^n\beta_k u_k\right\rangle\\ & = & \sum_{j=1}^n\left( \alpha_j\left\langle u_j, \sum_{k=1}^n\beta_k u_k\right\rangle\right). \end{eqnarray*} Then applying linearity on the second component you get the given expression.

share|improve this answer
    
Maybe it is even easier to see what is going on if you set $v_n= \langle v,u_j\rangle$ and $w_j=\langle w,u_j\rangle$? –  AD. Nov 2 '10 at 20:09
    
@AD: good point. –  Arturo Magidin Nov 2 '10 at 20:13
    
But don't put to much strength in alvoutila the user do not listen very well anyway. –  AD. Nov 2 '10 at 20:16

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.