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Math Help - Scalar product problem

  1. #1
    MHF Contributor arbolis's Avatar
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    Scalar product problem

    I feel sorry to ask help for this problem, since I believe it's a simple one. But I go nowhere when I think alone.
    The problem states :
    Let \langle \cdot ,\cdot\rangle be one of the following scalar product in the space of polynomials with real coefficients of degree \leq n.
    a) \langle f,g \rangle=\sum_{j=0}^n f(x_j)g(x_j), with x_i different from x_j when i different from j.
    b) \langle f,g \rangle=\int_0^1 f(x)g(x)dx.
    Prove that { 1,x,x^2,...,x^n} is not an orthogonal set if n\geq 2.
    --------------------------
    First : I don't know how to start. Second : in the item a), I don't understand why there is the condition of the "i", if it isn't even in the equation. Maybe an error of text, so it should be f(x_i)g(x_j)? Third : About what I must prove. It says, an orthogonal set, wouldn't it be an orthogonal basis? But even having said that, I still don't understand at all what they ask me to do. Please help me!
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by arbolis View Post
    I feel sorry to ask help for this problem, since I believe it's a simple one. But I go nowhere when I think alone.
    The problem states :
    Let \langle \cdot ,\cdot\rangle be one of the following scalar product in the space of polynomials with real coefficients of degree \leq n.
    a) \langle f,g \rangle=\sum_{j=0}^n f(x_j)g(x_j), with x_i different from x_j when i different from j.
    b) \langle f,g \rangle=\int_0^1 f(x)g(x)dx.
    Prove that { 1,x,x^2,...,x^n} is not an orthogonal set if n\geq 2.
    --------------------------
    First : I don't know how to start. Second : in the item a), I don't understand why there is the condition of the "i", if it isn't even in the equation. Maybe an error of text, so it should be f(x_i)g(x_j)? Third : About what I must prove. It says, an orthogonal set, wouldn't it be an orthogonal basis? But even having said that, I still don't understand at all what they ask me to do. Please help me!

    Two Vectors are orthogonal if there inner product is zero

    For b) I think your inner product should be

    <f,g>=\int_{-1}^{1}f(x)g(x)dx



    Then the vectors 1 \\\ x are orthogonal

    <1,x>=\int_{-1}^{1}1\cdot x dx=\frac{1}{2}x^2 \bigg|_{-1}^{1}=\frac{1}{2}(1^2-(-1)^2)=0

    but if you try

    <br />
<1,x^2>=\frac{2}{3} \ne 0 so they are not orthogonal.

    I hope this helps.
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  3. #3
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    Quote Originally Posted by TheEmptySet View Post
    For b) I think your inner product should be

    <f,g>=\int_{-1}^{1}f(x)g(x)dx
    .
    Why do you think so?
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  4. #4
    Moo
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    A Cute Angle Moo's Avatar
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    Quote Originally Posted by Isomorphism View Post
    Why do you think so?
    It wouldn't be orthogonal for any n, whereas the question asks for n>2 :/
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  5. #5
    MHF Contributor arbolis's Avatar
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    I hope this helps.
    Yes it does. But I insist, I think it's \int_0^1 f(x)g(x)dx. If the lower limit of the integral would be -1, then I must show that the product of 2 different functions is an odd function. It wouldn't be always true, but as you shown, if n=2 then it doesn't work. So it's not even worth looking for further n.
    I think I'm starting to understand the problem. So please answer me to :
    it says, an orthogonal set, wouldn't it be an orthogonal basis?
    Now for the a). If j=0, I get f(1)g(1). How can they want me to say it's equal to 0? There's no way to suppose this!
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  6. #6
    Behold, the power of SARDINES!
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    The set \{ 1,x,x^2,...x^n\} does not form a orthogonal basis for the innerproduct space P_n with respect to the innerproduct <f,g>=\int_{-1}^{1}f(x)g(x)dx

    Look for the Orthogonality property here

    Legendre polynomials - Wikipedia, the free encyclopedia

    and for the Gram-Schmidt process here

    GramÔ€“Schmidt process - Wikipedia, the free encyclopedia

    If you use the Gram-Schmidt process on the above set you will obtain an orthogonal set with respect to the inner product mentioned it turns out to be the Legendre polynomials.
    Last edited by TheEmptySet; June 7th 2008 at 03:57 PM. Reason: I left out an important word "orthogonal"
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  7. #7
    MHF Contributor arbolis's Avatar
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    Thanks a lot TheEmptySet. It looks very interesting to me even if quite hard to understand.
    And well... I give up for the a). I'll ask help to a friend on Monday, maybe he understood this.
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