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Abstract Vector Space Question

  1. Mar 4, 2012 #1
    1. The problem statement, all variables and given/known data
    Let [itex]g_1(t) = t - 1[/itex] and [itex]g_2(t)= t^2+t[/itex]. Using the inner product on [itex]P_2[/itex] defined in example 10(b) with [itex]t_1=-1,t_2=0,t_3=1[/itex], find a basis for the orthogonal complement of [itex]Span(g_1, g_2)[/itex].

    2. Relevant equations
    From example 10(b)
    [itex]\langle p, q \rangle = \sum_{i=1}^{k+1} p(t_i)q(t_i)[/itex]

    [itex](p,q)\in P_2[/itex]

    3. The attempt at a solution
    Well the orthogonal complement of [itex]span(g_1,g_2)[/itex] will be [itex]x[/itex] such that [itex]x\cdot (c_1g_1 + c_2g_2) = 0[/itex], but how can I find the basis for that set? And why and where do we need to use inner product? Im confused
    Last edited by a moderator: Mar 4, 2012
  2. jcsd
  3. Mar 4, 2012 #2


    Staff: Mentor

    Start with this equation. Here, x [itex]\in[/itex] P2, so you can write it as a + bt + ct2.

    Where you show the dot product in the equation above, you should be using the inner product from example 10b. Also, put in the formulas for the functions g1 and g2.

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