Linear algebra proof - Orthogonal complements

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Homework Statement



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.

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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.
 
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Answers and Replies

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Dick
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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?
 

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