# Gram-Schmidt Process on these Vectors

• Dec 6th 2010, 08:14 PM
Anthonny
Gram-Schmidt Process on these Vectors
I have a vector space of polynomials at most second degree with basis $(1,x,x^2)$ with the inner product defined as

$\langle{p,q}\rangle=\int^{10}_{0}p(x)q(x)dx$

and I want to find an orthonormal basis using the Gram-Schmidt process.

I've found that $e_1=1/10$

For $e_2$ I have

$e_2=(x-\langle{x},\frac{1}{10}\rangle\frac{1}{10})/||x-\langle{x},\frac{1}{10}\rangle\frac{1}{10}||$

$e_2=(x-1/2)/||x-1/2||$
$e_2=(x-1/2)/16.90661$

However when I take the inner product between $e_1$ and $e_2$ the result is not zero, but it should be since they're orthogonal. However if I take the inner product with the upper limit at 1 they are zero.

I'm pretty sure I've been applying the Gram-Schmidt process using the inner product I defined above. Does anyone know what I did wrong?
• Dec 6th 2010, 08:22 PM
dwsmith
$\displaystyle e_1=\frac{1}{\sqrt{10}}$
• Dec 6th 2010, 08:27 PM
Anthonny
Quote:

Originally Posted by dwsmith
$\displaystyle e_1=\frac{1}{\sqrt{10}}$

I cannot believe I overlooked the square root step even though I constantly reminded myself to take the square root.

Thank you very much!
• Dec 6th 2010, 08:38 PM
dwsmith
No problem.