# finding an orthonormal basis

• April 22nd 2008, 04:31 PM
mathisthebestpuzzle
finding an orthonormal basis
find an orthonormal basis of polynomials of degree 2 over the real space, with inner product <p,q> = integral from (0,1) of p(x)q(x)dx.

if you could direct me as to how to insert an integral symbol that would be great as well!
• April 22nd 2008, 09:53 PM
CaptainBlack
Quote:

Originally Posted by mathisthebestpuzzle
find an orthonormal basis of polynomials of degree 2 over the real space, with inner product <p,q> = integral from (0,1) of p(x)q(x)dx.

if you could direct me as to how to insert an integral symbol that would be great as well!

You start with a known basis say $\{1, x, x^2\}$ then apply the Gramm-Schmidt process to generate an orthogonal basis, and then normalise the new basis.

RonL

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• April 22nd 2008, 10:11 PM
mathisthebestpuzzle
can you explain the normalization of the orthonormal basis? i have found the gramm-schmidt applied to the 1, x, and x^2. what must i do to normalize?

ps. you're a baller.
• April 22nd 2008, 10:19 PM
Isomorphism
Quote:

Originally Posted by mathisthebestpuzzle
can you explain the normalization of the orthonormal basis? i have found the gramm-schmidt applied to the 1, x, and x^2. what must i do to normalize?

ps. you're a baller.

For any vector u, the normalised vector e is given by:
$\mathbf{e} = {\mathbf{u}\over \|\mathbf{u}\|}$
• April 22nd 2008, 10:23 PM
CaptainBlack
Quote:

Originally Posted by mathisthebestpuzzle
can you explain the normalization of the orthonormal basis? i have found the gramm-schmidt applied to the 1, x, and x^2. what must i do to normalize?

ps. you're a baller.

$u_1=1$ the constant function

$u_2=x - \frac{\langle x, u_1 \rangle}{\langle u_1, u_1 \rangle}u_1$

where $\langle u_1, u_1 \rangle=\int_0^1 1 ~dx =1$, and $\langle x, u_1 \rangle=\int_0^1 x ~dx =1/2$, so:

$
u_2=x-1/2
$

Now repeat to find $u_3$:

$u_2=x^2 - \frac{\langle x^2, u_1 \rangle}{\langle u_1, u_1 \rangle}u_1- \frac{\langle x^2, u_2 \rangle}{\langle u_2, u_2 \rangle}u_2$

$\{u_1, u_2, u_3 \}$ is an orthogonal basis, you now have to normalise them to get your orthonormal basis: $\{e_1, e_2, e_3 \}$, where:

$e_i=\frac{u_i}{\langle u_i, u_i \rangle^{1/2}}$

RonL