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Math Help - Orthogonal vector

  1. #1
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    Orthogonal vector

    Hey, I wrecked my head around this one and I hope you could help me.
    The questions is as follows:
    "Let V be an inner product space, and U,W be subspaces of V so that dim(U)<dim(W)
    Prove that exists a vector w in W (that is not the zero vector) so that it is orthogonal to all the vectors that are in U"

    Thank you in advance!
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  2. #2
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    Re: Orthogonal vector

    Hey zokomoko.

    Hint: Use the identity that if V < W (i.e. V is a subspace of W) then V + V_perp = W where V_perp is perpendicular to V.
    Thanks from zokomoko
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  3. #3
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    Re: Orthogonal vector

    Got it!
    Can't believe it was as simple as that
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  4. #4
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    Re: Orthogonal vector

    Another way to do this: choose a basis for U. Then extend it to a basis for W. Let v be any vector such that its linear combination of those basis vectors has coefficient 0 for vectors that are basis vectors for U, non-zero coefficient for at least one of the basis vectors in the extension.
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