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Gram-Schmidt Orthogonalization Process

  1. Apr 14, 2017 #1
    1. The problem statement, all variables and given/known data
    Find an orthogonal basis for ##\operatorname{span} (S)## if ##S= \{1,x,x^2 \}##, and ##\langle f,g \rangle = \int_0^1 f(x) g(x) \, dx##

    2. Relevant equations


    3. The attempt at a solution
    So we start by the normal procedure.

    Let ##v_1 = 1##. Then ##\displaystyle v_2 = x - \frac{\langle x,1 \rangle}{\| 1 \|^2}(1) = x - \frac{1}{2}##.
    Then ##\displaystyle v_3 = x^2 - \frac{\langle x^2,1 \rangle}{\| 1 \|^2}(1) - \frac{\langle x^2,x \rangle}{\| x \|^2}(x) = x^2 - \frac{1}{3} - \frac{3}{4}x##.

    But this is not correct, because if I calculate ##\displaystyle \langle 1, x^2 - \frac{1}{3} - \frac{3}{4}x\rangle = \int_0^1 x^2 - \frac{1}{3} - \frac{3}{4}x \, dx = -\frac{3}{2} \ne 0##.

    What am I doing wrong?
     
  2. jcsd
  3. Apr 14, 2017 #2

    FactChecker

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    Shouldn't the v3 definition use v2 = x-1/2 in the third term instead of x?
     
  4. Apr 15, 2017 #3

    Ray Vickson

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    Wrong normalization.
    $$v_2=\frac{u_2}{|| u_2 ||}, \\
    u_2 = x - <x, v_1> v_1 $$
     
    Last edited: Apr 15, 2017
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