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!

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- Apr 22nd 2008, 04:31 PMmathisthebestpuzzlefinding 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! - Apr 22nd 2008, 09:53 PMCaptainBlack
You start with a known basis say $\displaystyle \{1, x, x^2\}$ then apply the Gramm-Schmidt process to generate an orthogonal basis, and then normalise the new basis.

RonL

(To typeset mathematics this site uses LaTeX, see the tutorial here) - Apr 22nd 2008, 10:11 PMmathisthebestpuzzle
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. - Apr 22nd 2008, 10:19 PMIsomorphism
- Apr 22nd 2008, 10:23 PMCaptainBlack
$\displaystyle u_1=1$ the constant function

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

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

$\displaystyle

u_2=x-1/2

$

Now repeat to find $\displaystyle u_3$:

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

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

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

RonL