Lineaer algebra - orthogonality

1. Feb 8, 2013

yy205001

1. The problem statement, all variables and given/known data
Let<x,y> be an inner product on a vector space V, and let e1, e2,...,en be an orthonormal basis for V.
Prove: <x,y> = <x,e1><y,e1>+...+<x,en><y,en>.

2. Relevant equations

<x,x> = abs(x)
a<x,y> = <ax,y> = <x,ay>

3. The attempt at a solution

RHS
=<x,y>
= x1y1+...+xnyn
= x1<e1,e1>y1<e1,e1>+....+xn<en,en>yn<en,en>
* <e,e>=1
=e12<x1,e1>+...+en2<xn,en>

And i cannot find a way to take the e2 out of the equation.
Any help is appreciated

2. Feb 8, 2013

ehild

How is an orthonormal basis defined?

A subset {v_1,...,v_k} of a vector space V, with the inner product <,>, is called orthonormal if <v_i,v_j>=0 when i≠j. That is, the vectors are mutually perpendicular. Moreover, they are all required to have length one: <v_i,v_i>=1.

ehild

3. Feb 8, 2013

yy205001

Orthonormal basis
A set of vectors for an inner product space whose are linearly independence and span V and orthonormal.

orthonormal: orthogonal (<x,y>=0) and each vector has length one.

4. Feb 8, 2013

HallsofIvy

Staff Emeritus
Under "relevant equations" you have "<x, x>= abs(x)". There is NO "x2" so what, exactly do you mean by "en2"? <en, en>?
You also say "orthonormal: orthogonal (<x,y>=0) and each vector has length one."