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Homework Help: Linear algebra proof - Orthogonal complements

  1. Nov 15, 2009 #1
    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. jcsd
  3. Nov 15, 2009 #2


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    Homework Helper

    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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