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Math Help - Inner product/subspace help!

  1. #1
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    Inner product/subspace help!

    Hi, I have this problem and don't know how to go about it:

    (P2 being the set of all polynomials of degree less than or equal to 2)


    Let p(x), q(x) Є P2. You may assume that:

    <p(x), q(x)> = \int p(x)q(x)dx (with limits from -1 to 1) defines an inner product on P2.



    (A) Find a basis for the subspace of P2:
    V= {a+b+ax+bx^2|a,b Є R}

    (B) Using the inner product defined above and the basis vectors found in (A), use the Gram-Schmidt procedure to find an orthonormal basis for V.


    Thank you.
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  2. #2
    Senior Member roninpro's Avatar
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    What is it that you are having trouble with?
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  3. #3
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    Finding a basis for the subspace of P2, to get started, given that I have a rule: V= {a+b+ax+bx^2|a,b Є R}. No idea?
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  4. #4
    Senior Member roninpro's Avatar
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    Let's rearrange some of the terms in that set.

    \{a(x+1)+b(x^2+1)\ |\ a,b\in \mathbb{R}\}

    Can you see how to pick the basis now?
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  5. #5
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    I'm sorry,

    What do we do the procedure on? What inner product?
    What's the basis got to do with it?
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  6. #6
    Senior Member roninpro's Avatar
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    I'm just looking at part A of your problem. It just asks you to find a basis for your set. So far, it has nothing to do with inner product space or Gram-Schmidt.
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  7. #7
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    For (A), is the basis {(0,1,1),(1,0,1)}

    As in: x+1 = (0,1,1)
    x^2 + 1 = (1,0,1)
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  8. #8
    Senior Member roninpro's Avatar
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    Looks good. Now are you able to do Gram-Schmidt?
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