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  1. #1
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    Question finding an orthonormal basis

    find an orthonormal basis of polynomials of degree 2 over the real space, with inner product <p,q> = integral from (0,1) of p(x)q(x)dx.

    if you could direct me as to how to insert an integral symbol that would be great as well!
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  2. #2
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    Quote Originally Posted by mathisthebestpuzzle View Post
    find an orthonormal basis of polynomials of degree 2 over the real space, with inner product <p,q> = integral from (0,1) of p(x)q(x)dx.

    if you could direct me as to how to insert an integral symbol that would be great as well!
    You start with a known basis say \{1, x, x^2\} then apply the Gramm-Schmidt process to generate an orthogonal basis, and then normalise the new basis.

    RonL

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  3. #3
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    can you explain the normalization of the orthonormal basis? i have found the gramm-schmidt applied to the 1, x, and x^2. what must i do to normalize?

    ps. you're a baller.
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  4. #4
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    Quote Originally Posted by mathisthebestpuzzle View Post
    can you explain the normalization of the orthonormal basis? i have found the gramm-schmidt applied to the 1, x, and x^2. what must i do to normalize?

    ps. you're a baller.
    For any vector u, the normalised vector e is given by:
    \mathbf{e} = {\mathbf{u}\over \|\mathbf{u}\|}
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  5. #5
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    Quote Originally Posted by mathisthebestpuzzle View Post
    can you explain the normalization of the orthonormal basis? i have found the gramm-schmidt applied to the 1, x, and x^2. what must i do to normalize?

    ps. you're a baller.
    u_1=1 the constant function

    u_2=x - \frac{\langle x, u_1 \rangle}{\langle u_1, u_1 \rangle}u_1

    where \langle u_1, u_1 \rangle=\int_0^1 1 ~dx =1, and \langle x, u_1 \rangle=\int_0^1 x ~dx =1/2, so:

     <br />
u_2=x-1/2<br />

    Now repeat to find u_3:

    u_2=x^2 - \frac{\langle x^2, u_1 \rangle}{\langle u_1, u_1 \rangle}u_1- \frac{\langle x^2, u_2  \rangle}{\langle u_2, u_2 \rangle}u_2

    \{u_1, u_2, u_3 \} is an orthogonal basis, you now have to normalise them to get your orthonormal basis: \{e_1, e_2, e_3 \}, where:

    e_i=\frac{u_i}{\langle u_i, u_i \rangle^{1/2}}

    RonL
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