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Math Help - Finding an Orthonormal Basis

  1. #1
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    Finding an Orthonormal Basis



    The inner product here is defined by the integration of p(x)q(x) from 0 to 1.

    I sort of understand why this is true, but I can't prove it (so I guess I don't fully understand it).
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  2. #2
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    Quote Originally Posted by davismj View Post


    The inner product here is defined by the integration of p(x)q(x) from 0 to 1.

    I sort of understand why this is true, but I can't prove it (so I guess I don't fully understand it).

    Google "Gram-Schmidt orthogonal basis"

    Tonio
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  3. #3
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    I know how to do a Gram-Schmidt procedure. What I don't understand is how to show that the matrix for the differentiation operator on the orthonormal basis (1,sqrt(3)(2x-1),sqrt(5)(6x^2-6x+1)) is upper triangular...
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    Quote Originally Posted by davismj View Post
    I know how to do a Gram-Schmidt procedure. What I don't understand is how to show that the matrix for the differentiation operator on the orthonormal basis (1,sqrt(3)(2x-1),sqrt(5)(6x^2-6x+1)) is upper triangular...

    Why did you choose precisely that basis for  P_2[x]_{\mathbb{R}}? Check that wrt the basis \{1,x,x^2\}, the matrix of the diff. operator ALREADY is upper triangular, so just carry on the GS process on this basis...

    Tonio
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