Gram-Schmidt Orthogonalization Process

1. Apr 14, 2017

Mr Davis 97

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. Apr 14, 2017

FactChecker

Shouldn't the v3 definition use v2 = x-1/2 in the third term instead of x?

3. Apr 15, 2017

Ray Vickson

Wrong normalization.
$$v_2=\frac{u_2}{|| u_2 ||}, \\ u_2 = x - <x, v_1> v_1$$

Last edited: Apr 15, 2017