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Finding an orthogonal basis?

  1. Mar 23, 2009 #1
    1. The problem statement, all variables and given/known data

    Let P2 denote the space of polynomials in k[x] and degree < or = 2. Let f, g exist in P2 such that

    f(x) = a2x^2 + a1x + a0
    g(x) = b2x^2 + b1x + b0

    Define

    <f, g> = a0b0 + a1b1 + a2b2

    Let f1, f2, f3, f4 be given as below

    f1 = x^2 + 3
    f2 = 1 - x
    f3 = 2x^2 + x + 1
    f4 = x + 1

    Find an orthogonal basis of Span(f1, f2, f3, f4).


    2. Relevant equations

    Gram-Schmidt orthogonalization process.

    3. The attempt at a solution

    Span(f1, f2, f3, f4) = Span(w1, w2, w3, w4)

    Take S = {f, 1}

    w1 = f1
    w2 = 1 - (<1, f1>/<f1, f1>)*f1
    w3 = 1 - (<1, f2>/<f2, f2>)*f2 - (<1, f1>/<f1, f1>)*f1
    w4 = 1 - (<1, f3>/<f3, f3>)*f3 - (<1, f2>/<f2, f2>)*f2 - (<1, f1>/<f1, f1>)*f1

    Is this the correct procedure? Can I take g = 1 like that?
     
    Last edited: Mar 23, 2009
  2. jcsd
  3. Mar 23, 2009 #2

    Mark44

    Staff: Mentor

    Looks good to me. There's nothing wrong with starting with g(x) = 1.
     
  4. Mar 23, 2009 #3
    Thanks a bunch!
     
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