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Complementary polynomial space

  1. Apr 10, 2007 #1
    Let [tex] V = P^{(4)}[/tex] denote the space of quartic polynomials, with the L^2 inner product

    [tex]<p,q>= \int_{-l}^l p(x)q(x)dx[/tex]

    Let [tex] W = P^2[/tex] be the subspace of quadratic polynomials.
    a) Write down the conditions that a polynomial [tex] p \in P^{(4)}[/tex] must satisfy in order to belong to the orthogonal complement Wperp.
    b) Find a basis for and the dimension of Wperp.
    c) Find an orthogonal basis for Wperp.

    The first part just goes off the definition of a complementary subspace.
    [tex]W_{perp} = (q(x) = a + bx + cx^2 + dx^3 + ex^4 | <p,1>=<p,x>=<p,x^2>=0)[/tex]

    It looks like the second part wants me to actually do the calculation, but that looks like a lot of work to multiply it out. Is that really what it is asking?

    The last part should be easy, because I can just take the basis from part b) and apply the Gram-Schmidt method.
    Last edited: Apr 10, 2007
  2. jcsd
  3. Apr 11, 2007 #2


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    What is 'a lot of work to multiply out'? It's just monomials times a polynomial integrated over [-1,1]. You get three conditions on a,b,c,d,e. It doesn't get too much more straightforward than that.
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