Orthogonal Complements of a Subspace of Polynomials (LaTex notation)

• Dec 1st 2011, 07:53 PM
maxgunn555
Orthogonal Complements of a Subspace of Polynomials (LaTex notation)
$Let V be img.top {vertical-align:15%;} $R_{2}[x]$, and let S be the subspace of all polynomials p(x) satisfying

$\\ \frac{\2d^2\p}{dx^2} = p(2)$

Find the orthogonal complement of S with respect to the inner product

<p,q> = p(-1)q(-1) +p(0)q(0) + p(1)q(1)

And find the polynomial in S which is closest to 1 + x _ x^2

Thanks for any help.

" alt=" Let V be $R_{2}[x]$, and let S be the subspace of all polynomials p(x) satisfying

$\\ \frac{\2d^2\p}{dx^2} = p(2)$

Find the orthogonal complement of S with respect to the inner product

<p,q> = p(-1)q(-1) +p(0)q(0) + p(1)q(1)

And find the polynomial in S which is closest to 1 + x _ x^2

Thanks for any help.

" />
• Dec 1st 2011, 08:05 PM
Deveno
Re: Orthogonal Complements of a Subspace of Polynomials (LaTex notation)
• Dec 1st 2011, 08:44 PM
Drexel28
Re: Orthogonal Complements of a Subspace of Polynomials (LaTex notation)
Quote:

Originally Posted by maxgunn555
Let $V$ be $\mathbb{R}_{2}[x]$, and let $S$ be the subspace of all polynomials $p(x)$ satisfying

$\displaystyle \frac{d^2p}{dx^2} = p(2)$

Find the orthogonal complement of $S$ with respect to the inner product

$\langle p,q\rangle = p(-1)q(-1) +p(0)q(0) + p(1)q(1)$

And find the polynomial in $S$ which is closest to $1 + x + x^2$

Thanks for any help.

I believe this is the question. I now feel like you should show some work. Tell us where you are stuck, because this is a computation. So, what part of the setup don't you understand?
• Dec 1st 2011, 09:07 PM
maxgunn555
Re: Orthogonal Complements of a Subspace of Polynomials (LaTex notation)
Hi again, i attempted it a few days ago before moving onto other modules i have. but i know the orthogonal complement is a space where each vector is orthogonal to each vector in the subspace S. orthogonal meaning the inner product of two vectors if orthogonal to each other equal zero (or pi).
• Dec 1st 2011, 09:11 PM
Drexel28
Re: Orthogonal Complements of a Subspace of Polynomials (LaTex notation)
Quote:

Originally Posted by maxgunn555
Hi again, i attempted it a few days ago before moving onto other modules i have. but i know the orthogonal complement is a space where each vector is orthogonal to each vector in the subspace S. orthogonal meaning the inner product of two vectors if orthogonal to each other equal zero (or pi).

Right, so you have an equation. A polynomial $p(x)\in S^\perp$ if and only if $p(-1)q(-1)+p(0)q(0)+p(1)q(1)=0$ for every $q(x)=ax^2+bx+c$ such that $4a=4a+2b+c$, right>