Orthonormal Basis Homework: Gram-Schmidt Process w/ Inner Product

[maggie56]

Homework Statement

Hi, i am applying the gram-schmidt procedure to a basis of {1,2x,3x^2} with inner product <p,q> = $$\int p(x)q(x)$$ from 0 to 1.

i am unsure what to do with the inner product

Homework Equations

The Attempt at a Solution

I have followed the procedure i have for converting this basis to an orthonormal basis, where v1=1, v2=2x, v3=3x^2 are the initial vectors

i let b1=v1=1

so b2 = v2 + $$\alpha$$ v1 = 2x + $$\alpha$$
then <b1,b2> = 0 = <1, 2x + $$\alpha$$ >
which gives me $$\alpha$$ = -2x but then b2=0
I get a result of 0 for b3 also, and think i must have something wrong because i haven't used the inner product <p,q> = $$\int p(x)q(x)$$ from 0 to 1. but can't see what to do with this.


thanks

 

[lanedance]

ok so you want the inner product to be zero, so the vectors orthogonal
$$<b1,b2> = 0 = <1, 2x + \alpha > =\int_0^1 dx 2x+\alpha = (x^2 + \alpha.x)|_0^1$$

so this is the condition for alpha

notes alpha is a scalar multiplier of the basis vectors and cannot be a function of x.