Normalizing an Orthogonal Basis: A Step-by-Step Guide

[beetle2]

Homework Statement

I have used the gram schmidt process to find an orthogonal basis for ${1,t,t^2}$
which is
$(1,x,x^2 - \frac{2}{3})$

How to i normalize these

Homework Equations

$e_1=\frac{u_1}{|u_1|}$

The Attempt at a Solution

$e_1=\frac{1}{\sqrt{\int_{-1}^{1}f(1)g(1)}}=\frac{1}{\sqrt{2}}$


$e_2=\frac{x}{\sqrt{\int_{-1}^{1}f(x)g(x)}}=\frac{x}{\sqrt{\frac{2}{3}}}$

$e_3=\frac{x}{\sqrt{\int_{-1}^{1}f(x^2-3)g(x^2-3)}}=\frac{x^2}{\sqrt{\frac{-62}{45}}}$

 

[foxjwill]

Just divide each by its norm.

 

[Mark44]

There is a problem with your third function. It is orthogonal to x, but not to 1. Each function has to be orthogonal to each other function in your set.
$$\int_{-1}^1 1 (x^2 - 2/3) dx~=~\left[\frac{x^3}{3} - (2/3)x\right]_{-1}^1~=~ -2/3$$

Also, is the function x^2 - 2/3 or x^2 - 3? You are using both in your calculations.

 

[Mark44]

One other thing. The norm of a function is the square root of the inner product of it and itself.

$e_2=\frac{x}{\sqrt{\int_{-1}^{1}f(x)g(x)}}=\frac{x}{\sqrt{\frac{2}{3}}}$

You shouldn't have f(x) and g(x) in there for <x, x>, since f(x) = g(x) = x. It should be like this:
$e_2=\frac{x}{\sqrt{\int_{-1}^{1}x*xdx}}= ...$

 

[beetle2]

thanks your right my u3 should be $x^2-\frac{1}{3}$
which makes
$e_3=\frac{x^{2}-\frac{1}{3}}{\sqrt{\int_{-1}^{1}x^{2}-\frac{1}{3}*x^{2}-\frac{1}{3}dx}}= ...$

= $\frac{x^2-\frac{1}{3}}{\sqrt{frac{8}{45}}}$

hows that look

The innner product of $x^2-3 and \frac{1}{\sqrt{2}} = 0$ so they're orthogonal

 

[Mark44]

Much better.

For the quantity in the radical, you left off the \ before frac in your LaTeX code. That's why it looks like it does. Should look like this:
$\frac{x^2-\frac{1}{3}}{\sqrt{\frac{8}{45}}}$

 

[beetle2]

Thanks for your help guys much appreciated