- #1

yy205001

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## Homework Statement

Let<

**x**,

**y**> be an inner product on a vector space

**V**, and let e

_{1}, e

_{2},...,e

_{n}be an orthonormal basis for

**V**.

Prove: <x,y> = <x,e

_{1}><y,e

_{1}>+...+<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>

= x

_{1}y

_{1}+...+x

_{n}y

_{n}

= x

_{1}<e

_{1},e

_{1}>y

_{1}<e

_{1},e

_{1}>+...+x

_{n}<e

_{n},e

_{n}>y

_{n}<e

_{n},e

_{n}>

* <e,e>=1

=e

_{1}

^{2}<x

_{1},e

_{1}>+...+e

_{n}

^{2}<x

_{n},e

_{n}>

And i cannot find a way to take the e

^{2}out of the equation.

Any help is appreciated