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Math Help - gram-schmidt process help

  1. #1
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    gram-schmidt process help

    Hi everyone,

    i need your help with solving the (ii) choice of the quesiton i attached,thanks for your helps .
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  2. #2
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    Re: gram-schmidt process help

    by the way i confused how to use (u,v) and p(t),q(t),
    i really appreciate if you show me the details of obtaining g1 and g2 i can find g3.
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  3. #3
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    Re: gram-schmidt process help

    ok, since we can pick any basis vector to start with, i'll choose g1 = b1 = 1 (the constant polynomial, not the number) that is: b1(t) = 1, for all t.

    now to find g2, we have the find the projection of b2 in the direction of 1. we have to use the given inner product for this, but the general formula is:

    projg1(b2) = [(g1,b2)/(g1,g1)]g1.

    note that g1(0) = g1(1) = g1(2) = 1 (it's a constant function). thus (g1,g1) = (1)(1) + (1)(1) + (1)(1) = 3.

    the real action comes in when we compute (g1,b2). b2(t) = t, so:

    b2(0) = 0, b2(1) = 1, b2(2) = 2.

    thus (g1,b2) = g1(0)b2(0) + g1(1)b2(1) + g1(2)b2(2) = (1)(0) + (1)(1) + (1)(2) = 3.

    hence projg1(b2) = (3/3)(1) = 1 (the constant polynomial, not the number).

    to find g2, we simply subtract the projection of b2 in the direction of g1 from b2:

    g2(t) = b2(t) - projg1(b2)(t) = t - 1.

    let's verify that g2 is orthogonal to g1:

    (g1,g2) = (1,t-1) = g1(0)g2(0) + g1(1)g2(1) + g1(2)g2(2) = (1)(-1) + (1)(0) + (1)(1) = 0. see?

    finding g3 will be the same, except now you have to find 2 projections, and subtract them both from b3(t) = t2.
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