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Direct Sum Problem

  1. Oct 25, 2008 #1
    1. The problem statement, all variables and given/known data
    Let Pn denote the vector space of polynomials of degree less than or equal to n, and of the form p(x)=p0+p1x+...+pnxn, where the coefficients pi are all real. Let PE denote the subspace of all even polynomials in Pn, i.e., those that satisfy the property p(-x)=p(x). Similarly, let PO denote the subspace of all odd polynomials, i.e., those satisfying p(-x)=p(x). Show that Pn=PE[tex]\oplus[/tex]PO.


    2. Relevant equations
    Conditions for direct sum.


    3. The attempt at a solution

    PE=p1x+...+pn-1xn-1
    PO=p0+p2x2+...+pnxn
    such that n[tex]\in[/tex]{even real numbers}
    therefore, PE[tex]\bigcap[/tex]PO={[tex]\phi[/tex]}
    and PE+PO=p0+p1x+p2x2+...+pn-1+xn-1+pnxn=Pn

    is this sufficient to show that PE[tex]\oplus[/tex]PO=Pn ?

    I'm not so sure of what i have done and i know that my notations may be faulty, help please
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Oct 25, 2008 #2

    Dick

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    Science Advisor
    Homework Helper

    Yes, that works. But the intersection of PE and PO isn't {phi}. It's {0}. You've shown that everything in Pn can be written as a sum of something from PE and something from PO. I think the other thing you need to show is that those choices are unique.
     
    Last edited: Oct 25, 2008
  4. Oct 25, 2008 #3

    Mark44

    Staff: Mentor

    You have the even and odd polynomials switched.
    I would represent PE as p0 + p1x + p2x^2 + ... + p2nx^2n
    and PO as p1x + p3x^3 + ... + p2n-1x^(2n-1), in both cases where n is in the nonnegative integers.

    To show that Pn = PE [tex]\oplus[/tex] PO, don't you have to show that any arbitrary polynomial in Pn can be written as the sum of one polynomial from PE and another from PO? I'm a little rusty on this, so there might be some more that you have to show.
     
  5. Oct 25, 2008 #4

    Dick

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    Science Advisor
    Homework Helper

    Yes there is, you want to show that this decomposition is unique. E.g. Pn=Pn+PE. But that's not a direct sum.
     
  6. Oct 26, 2008 #5

    I figured i switched them, i guess i was a litle sleepy while typing. Thanks a lot though
     
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