Lineaer algebra - orthogonality

[yy205001]

Homework Statement

Let<x,y> be an inner product on a vector space V, and let e₁, e₂,...,e_n be an orthonormal basis for V.
Prove: <x,y> = <x,e₁><y,e₁>+...+<x,e_n><y,e_n>.

Homework Equations

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

The Attempt at a Solution

RHS
=<x,y>
= x₁y₁+...+x_ny_n
= x₁<e₁,e₁>y₁<e₁,e₁>+...+x_n<e_n,e_n>y_n<e_n,e_n>
* <e,e>=1
=e₁²<x₁,e₁>+...+e_n²<x_n,e_n>

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

 

[ehild]

How is an orthonormal basis defined?

Spoiler

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

 

[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.

 

[HallsofIvy]

Under "relevant equations" you have "<x, x>= abs(x)". There is NO "x²" so what, exactly do you mean by "e_n²"? <e_n, e_n>?
You also say "orthonormal: orthogonal (<x,y>=0) and each vector has length one."