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Math Help - Gram-Schmidt Process Calculation

  1. #1
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    Gram-Schmidt Process Calculation

    Let P3 be the four-dimensional vector space of all polynomials of degree ≤ 3 with inner product

    <f,g> = integral of f(x)g(x).dx from -1 to 1.

    Apply Gram-Schmidt Process to basis {1, x, x2, x3} of P3 to obtain an orthonormal basis {z1, z2, z3, z4} for this inner product on P3.

    I got the following but the numbers look extremely weird so I was wondering if others got the same answers:
    z1 = 1/sqrt(2),
    z2 = (3/sqrt(6))x,
    z3 = (15/(2sqrt(10)))(x^2 - 1/3)
    z4 = (35/(2sqrt(14)))(x^3 - (3/5)x)
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  2. #2
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    Re: Gram-Schmidt Process Calculation

    It's easy to check, isn't it?
    <z1, z1>=\int_{-1}^1 \frac{1}{2}dx= \left[\frac{x}{2}\right]_{-1}^1= 1
    <z2, z2>= \int_{-1}^1 \frac{9}{6}x^2dx= \left[\frac{1}{2}x^3\right]_{-1}^1= 1
    <z3, z3>= \int_{-1}^1 \frac{225}{40} (x^4- (2/3)x^2+ 1/9)dx= \left[\frac{45}{8}(x^5/5- (2/9)x^3+ x/9)\right]_{-1}^1= \frac{45}{8}(\frac{2}{5}- \frac{4}{9}+ \frac{2}{9})= 1
    etc.
    Thanks from twilightmage13
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  3. #3
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    Re: Gram-Schmidt Process Calculation

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