# Homework Help: Linear algebra proof - Orthogonal complements

1. Nov 15, 2009

### reb659

1. The problem statement, all variables and given/known data

Let V be an inner product space, and let W be a finite dimensional subspace of V. If x is not an element of W, prove that there exists y in V such that y is in the orthogonal complement of W, but the inner product of x and y is not equal to 0.

2. Relevant equations

3. The attempt at a solution

I'm pretty lost. There is a theorem which states:

Let W be a finite dimensional subspace of an inner product space V, and let y be in V. Then there exist unique vectors u in W andf z in the orthogonal complement of W such that y=u+z.

But I don't know how this would apply.

Last edited: Nov 15, 2009
2. Nov 15, 2009

### Dick

You can use that. Call U the orthogonal complement of W. Then your vector can be split into x=w+u where w is in W and u is in U. If x is not in W then u is not zero, agree with that? The inner product of u with u is then nonzero. Do you see it now?