- #1

ElijahRockers

Gold Member

- 270

- 10

## Homework Statement

Let ##V## be an inner product space and let ##V_0## be a finite dimensional subspace of ##V##. Show that if ##v ∈ V## has ##v_0 = proj_{V_0}(v)##:

||v - vo||^2 = ||v||^2 - ||vo||^2

## Homework Equations

General inner product space properties, I believe.

## The Attempt at a Solution

So Vo is finite dimensional and has an orthonormal basis {v1, v2,...., vn} for finite n. I can say that right? If so,

##v_0 = \sum_{i=1}^{n}<v,v_i>v_i## (Not sure if that helps really)

Also <v - vo , vo> = 0 since they are orthogonal.

That's about where I get stuck. I see that the expression I'm supposed to arrive at is basically pythagorean theorem for inner product spaces, and I can see it in my head quite easily if n is 2 dimensional.

So I tried working backwards:

||v - vo||^2 = ||v||^2 - ||vo||^2

||v - vo||^2 + ||vo||^2 = ||v||^2

And using inner product properties I can get:

<v - vo , v - vo> + <vo, vo> = <v , v>

I don't know if that really helps though.

Any advice is appreciated.