# Gram-Schmidt Process Calculation

• Mar 30th 2013, 11:38 AM
twilightmage13
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)
• Mar 30th 2013, 12:33 PM
HallsofIvy
Re: Gram-Schmidt Process Calculation
It's easy to check, isn't it?
$\displaystyle <z1, z1>=\int_{-1}^1 \frac{1}{2}dx= \left[\frac{x}{2}\right]_{-1}^1= 1$
$\displaystyle <z2, z2>= \int_{-1}^1 \frac{9}{6}x^2dx= \left[\frac{1}{2}x^3\right]_{-1}^1= 1$
$\displaystyle <z3, z3>= \int_{-1}^1 \frac{225}{40} (x^4- (2/3)x^2+ 1/9)dx= \left[\frac{45}{8}(x^5/5- (2/9)x^3+ x/9)\right]_{-1}^1= \frac{45}{8}(\frac{2}{5}- \frac{4}{9}+ \frac{2}{9})= 1$
etc.
• Mar 30th 2013, 11:37 PM
vicwilson123
Re: Gram-Schmidt Process Calculation