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Orthonormal Set spanning the subspace (polynomials)

  1. Oct 7, 2014 #1
    1. The problem statement, all variables and given/known data
    In the linear space of all real polynomials with inner product (x, y) = integral (0 to 1)(x(t)y(t))dt, let xn(t) = tn for n = 0, 1, 2,.... Prove that the functions y0(t) = 1, y1(t) = sqrt(3)(2t-1), and y2 = sqrt(5)(6t2-6t+1) form an orthonormal set spanning the same subspace as {x0, x1, x2}.

    2. Relevant equations


    3. The attempt at a solution
    I was attempting to use the Legendre Polynomials Rules to show that these polynomials form the basis but when I devise the Legendre Polynomials, they are different than those given.
     
  2. jcsd
  3. Oct 7, 2014 #2

    LCKurtz

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    What's stopping you? Can you prove they are orthogonal? Orthonormal? Span the same space?
     
  4. Oct 7, 2014 #3

    vela

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    What are the "Legendre Polynomials Rules"?
     
  5. Oct 8, 2014 #4

    HallsofIvy

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    So [itex]x_0= 1[/itex], [itex]x_1= t[/itex], and [itex]x_2= t^2[/itex]. What subspace do those span?

    [itex]y_0(t) = 1[/itex], [itex]y_1(t) = \sqrt{3}(2t-1)[/itex], and [itex]y_2 = \sqrt{5}(6t^2-6t+1)[/itex]. Show that these span the same subspace as the above. To show that they are orthonormal (which would also show that they are independent) you need to do 6 integrals.

    For example, [itex]\int_0^1 (y_0(t))^2 dt[/itex] must be equal to 1 while [itex]\int_0^1 y_0y_1 dt[/itex] must be equal to 0.
     
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