# Inner product proof

1. Nov 11, 2009

### jbear12

Let V be an inner product space, and let W be a finite-dimensional subspace of V. If x$$\notin$$ W, prove that there exists y$$\in$$ V such that y $$\in$$ W(perp), but <x,y>$$\neq$$ 0.

I don't have a clue....
Thanks

2. Nov 11, 2009

### lanedance

you could start by using something along the lines of gram-schmidt to decompose to x into a sum of vector from W & one from W perp...

3. Nov 12, 2009

### jbear12

Umm..I dont really get it. Can you explain more specifically? Thank you.

4. Nov 12, 2009

### lanedance

what don't you get?

first you need to assume x is non-zero

x is not contained in W, and as its non-zero, this means it must have a component in W perp , (as V = W + W perp by defintion of W perp, sloppy notation here, but hopefully you get the idea)

now consider the dot product of x with the component of x in W perp