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**kaelbu** Problem: A) Let $\displaystyle \{v_1, v_2, ..., v_3\}$ be an orthonormal basis for V. For any $\displaystyle x,y \epsilon V$ prove that

$\displaystyle <x,y>= \sum\limits_{i=1}^n <x,v_i> \overline{<y,v_i>} $

B) use (A) to prove that if $\displaystyle \beta$ is an orthonormal basis for V with inner product $\displaystyle <*, *>$ then for any $\displaystyle x,y\epsilon V$ $\displaystyle <\phi_\beta (x),\phi_\beta (y)>' = <[x]_\beta.[y]_\beta>'=<x,y>$ where $\displaystyle <*, *>$ is the standard product on $\displaystyle F^n$

Thoughts:

I worked out (A) but I am almost certain I did it wrong, however, this is what I did:

$\displaystyle <x,y>= \sum\limits_{i=1}^n <x,v_i> \overline{<y,v_i>} = \sum\limits_{i=1}^n <x,v_i> <v_i,y> $

$\displaystyle = \sum\limits_{i=1}^n\sum\limits_{j=1}^n x_j y_{ij} v_{ij} y_j = \sum\limits_{i=1}^n\sum\limits_{j=1}^n x_j y_j =<x,y>$

And I haven't the slightest idea what (B) means, let alone how to prove it using A which I almost certainly did wrong.