# Orthonormal basis

• Aug 30th 2010, 03:27 AM
Orthonormal basis
Define an inner product on P2 by <p,q> = ʃ (-1 to 1) p(x)q(x) dx. Let U (element of) P2 be the subspace spanned by {x,x^2}

(a) Find an orthonormal basis for U
(b) Find the polynomial in U that is as close as possible to f(x)=1 (for the norm corresponding to the above inner product).

Can someone help me solve this question? Thank you in advance.(Thinking)
• Aug 30th 2010, 04:04 AM
CaptainBlack
Quote:

Originally Posted by ghadz7
Define an inner product on P2 by <p,q> = ʃ (-1 to 1) p(x)q(x) dx. Let U (element of) P2 be the subspace spanned by {x,x^2}

(a) Find an orthonormal basis for U

Use the Gramm-Schmidt process on $\displaystyle \{x,x^2\}$

Quote:

(b) Find the polynomial in U that is as close as possible to f(x)=1 (for the norm corresponding to the above inner product).

Once you have the orthonormal basis elements $\displaystyle \{p_1(x), p_2(x)\}$ the polynomial is:

$\displaystyle P(x)=\langle f,p_1\rangle p_1(x)+\langle f,p_2\rangle p_2(x)$

CB
• Aug 30th 2010, 04:10 AM