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Math Help - Orthonormal Vector Property

  1. #1
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    Orthonormal Vector Property

    This problem comes from a second-semester course in linear algebra. We are currently covering Gram Scmidt - but outside of the application problems, I'm a bit lost.

    Problem: Suppose V is an inner product space and {v_1, ... , v_k} is an orthonormal set of vectors in V. Show that for all v in V, we have:
    \sum_{i\ =\ 0}^k |<v,v_j>|^2 \leq ||v||^2.

    Furthermore, show that the equality holds if and only if v is in the span of {v_1, ... , v_k}.

    Intuition: I can't picture the problem in anything larger than 2-space. If in two space, it seems plausible that a right triangle could be formed using the orthonormal set and v_j (which would be the hypotenuse). I'm not sure if this stems from a theorem, or if I should just be playing with the definitions in the summation/inequality.
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  2. #2
    Grand Panjandrum
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    Re: Orthonormal Vector Property

    Quote Originally Posted by jsndacruz View Post
    This problem comes from a second-semester course in linear algebra. We are currently covering Gram Scmidt - but outside of the application problems, I'm a bit lost.

    Problem: Suppose V is an inner product space and {v_1, ... , v_k} is an orthonormal set of vectors in V. Show that for all v in V, we have:
    \sum_{i\ =\ 0}^k |<v,v_j>|^2 \leq ||v||^2.

    Furthermore, show that the equality holds if and only if v is in the span of {v_1, ... , v_k}.

    Intuition: I can't picture the problem in anything larger than 2-space. If in two space, it seems plausible that a right triangle could be formed using the orthonormal set and v_j (which would be the hypotenuse). I'm not sure if this stems from a theorem, or if I should just be playing with the definitions in the summation/inequality.
    Write v=v'+v'' where v'\in {\rm{Span}}(v_1, .. v_k) and v'' is in the orthogonal complement of {\rm{Span}}(v_1, .. v_k)

    CB
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