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Polynomials and the Inner Product

  1. Dec 11, 2006 #1
    The question requires me to check whether the following formulae satisfy the properties of an inner product given the linear space of all real polynomials.
    [tex]
    f(1)g(1)[/tex]

    [tex]\left(\int_{0}^{1}f(t)dt\right)\left(\int_{0}^{1}g(t)dt\right)[/tex]

    The properties are satisfied in both cases (at least, that's my answer), but the book says 'No'. How could this be?
     
    Last edited: Dec 11, 2006
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  3. Dec 11, 2006 #2

    arildno

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    Eeh, wherever did you get those integrals from????
    You have been GIVEN that the formula is (f,g)=f(1)*g(1).
    THAT is the formula you are asked to check whether is correct or not!
     
  4. Dec 11, 2006 #3

    cristo

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    I think these might be two different questions, since the integrals don't follow from the first expression!!
     
  5. Dec 11, 2006 #4
    Sorry...cristo's right. Edited the post.
     
  6. Dec 11, 2006 #5

    AKG

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    Show some work. What makes you think that the properties are satisfied?
     
  7. Dec 11, 2006 #6
    Okay, for the first one...
    Under the usual rules of multiplication...

    f(1)g(1) = g(1)f(1)
    (cf)(1)g(1) = c(f(1)g(1))
    f(1)(g+h)(1) = f(1)g(1) + f(1)h(1)
    f(1)f(1) = (f(1))2 > 0 for f(1) != 0, and of course, it's zero when f(1) = 0.

    Second...
    int(f)int(g) = int(g)int(f)
    int(f)int(g+h) = int(f)( int(g)+int(h)) = int(f)(int(g)+int(f)int(h)
    int(cf)int(g) = c int(f)int(g)
    int(f)int(f) = (int(f))^2 >= 0
     
  8. Dec 11, 2006 #7

    mathwonk

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    whatb ARE the properties of an inner product? MAYBE ONE OF THEM IS SUPPOSED TO BE positive definiteness.
     
  9. Dec 11, 2006 #8
    Commutativity, Distributivity, Associativity and Positivity(that's the name given in the book for the last property).

    I'm afraid I don't understand. :uhh:
     
  10. Dec 11, 2006 #9

    NateTG

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    What happens to the 4th condition if:
    [tex]f(x)=\frac{1}{2}-x[/tex]
     
  11. Dec 11, 2006 #10

    AKG

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    What is positivity?
     
  12. Dec 11, 2006 #11
    f(1)f(1) = -0.5x(-0.5) = 0.25>0 ?
     
  13. Dec 11, 2006 #12

    AKG

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    He meant in relation to the integral inner product.
     
  14. Dec 11, 2006 #13
    Ah! Okay, I get it. f(x) is not zero, but the inner product <f,f> is. Therefore it does not satisfy the last condition.

    Similarly f(x) = x-1 is an example where the last condition is not satisfied (for the first problem), right?
     
  15. Dec 11, 2006 #14

    AKG

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    Right. When mathwonk mentioned "positive definiteness", he was referring to the following property:

    <f,f> > 0 if f is not the zero polynomial, and <f,f> = 0 if f is the zero polynomial.

    It is standard to call this property "positive definiteness", although you appear to be calling it "positivity".
     
  16. Dec 11, 2006 #15
    Thanks for the help, guys. :)

    Not me, but Tom M.Apostol. :tongue2:
     
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