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Math Help - Orthogonal vectors

  1. #1
    MHF Contributor alexmahone's Avatar
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    Orthogonal vectors

    Find a vector (w_1,w_2,w_3,...) that is orthogonal to v=(1,\frac{1}{2},\frac{1}{4},...). Compute its length ||w||.
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    Quote Originally Posted by alexmahone View Post
    Find a vector (w_1,w_2,w_3,...) that is orthogonal to v=(1,\frac{1}{2},\frac{1}{4},...). Compute its length ||w||.

    Orthogonal...in what vector space and with respect to what inner product??

    Tonio
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  3. #3
    MHF Contributor alexmahone's Avatar
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    Quote Originally Posted by tonio View Post
    Orthogonal...in what vector space and with respect to what inner product??

    Tonio
    Hilbert space with respect to dot product.
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    Quote Originally Posted by alexmahone View Post
    Hilbert space with respect to dot product.


    There are infinite Hilbert spaces ( to add "with inner product" is futile: ANY Hilbert space is a vector space with an inner product). I can think of at least

    two different Hilbert spaces with their inner products to which your vector belongs, so again: what vector space with what iiner product?

    Tonio
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    MHF Contributor alexmahone's Avatar
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    Quote Originally Posted by tonio View Post
    There are infinite Hilbert spaces ( to add "with inner product" is futile: ANY Hilbert space is a vector space with an inner product). I can think of at least

    two different Hilbert spaces with their inner products to which your vector belongs, so again: what vector space with what iiner product?

    Tonio
    I think the question merely requires us to find w_1,w_2,w_3,... such that w_1+\frac{w_2}{2}+\frac{w_3}{4}+...=0.
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    Quote Originally Posted by alexmahone View Post
    I think the question merely requires us to find w_1,w_2,w_3,... such that w_1+\frac{w_2}{2}+\frac{w_3}{4}+...=0.

    That's what I also thought, but then you already know what to look for......

    Idea: \sum\limits^\infty_{n=0}\frac{1}{4^n}=\frac{4}{3} \, , \, \, \,\sum\limits^\infty_{n=1}\frac{1}{2^{2n-1}}=2\sum\limits^\infty_{n=1}\frac{1}{4^n}=2\cdot \frac{1}{3}=\frac{2}{3}

    Tonio
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  7. #7
    MHF Contributor alexmahone's Avatar
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    Quote Originally Posted by tonio View Post
    That's what I also thought, but then you already know what to look for......

    Idea: \sum\limits^\infty_{n=0}\frac{1}{4^n}=\frac{4}{3} \, , \, \, \,\sum\limits^\infty_{n=1}\frac{1}{2^{2n-1}}=2\sum\limits^\infty_{n=1}\frac{1}{4^n}=2\cdot \frac{1}{3}=\frac{2}{3}

    Tonio
    I'm afraid I don't follow your hint.
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    Quote Originally Posted by alexmahone View Post
    I think the question merely requires us to find w_1,w_2,w_3,... such that w_1+\frac{w_2}{2}+\frac{w_3}{4}+...=0.
    You also need to ensure that \sum|w_n|^2<\infty, and in fact you need to be able to compute that sum in order to do the second part of the question.

    Big hint: Can you find a solution in which only the first two coordinates of w are nonzero?
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  9. #9
    MHF Contributor alexmahone's Avatar
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    Quote Originally Posted by Opalg View Post
    You also need to ensure that \sum|w_n|^2<\infty, and in fact you need to be able to compute that sum in order to do the second part of the question.

    Big hint: Can you find a solution in which only the first two coordinates of w are nonzero?
    Thanks. w=(1,-2,0,...) and its length is \sqrt{5}.
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