Thread: Orthogonal Complements of a Subspace of Polynomials (LaTex notation)

1. Orthogonal Complements of a Subspace of Polynomials (LaTex notation)

$\displaystyle Let V be$\displaystyle R_{2}[x] $, and let S be the subspace of all polynomials p(x) satisfying$\displaystyle \\ \frac{\2d^2\p}{dx^2} = p(2)$Find the orthogonal complement of S with respect to the inner product <p,q> = p(-1)q(-1) +p(0)q(0) + p(1)q(1) And find the polynomial in S which is closest to 1 + x _ x^2 Thanks for any help.$

3. Re: Orthogonal Complements of a Subspace of Polynomials (LaTex notation)

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

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

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

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

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

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?

4. 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).

5. Re: Orthogonal Complements of a Subspace of Polynomials (LaTex notation)

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 $\displaystyle p(x)\in S^\perp$ if and only if $\displaystyle p(-1)q(-1)+p(0)q(0)+p(1)q(1)=0$ for every $\displaystyle q(x)=ax^2+bx+c$ such that $\displaystyle 4a=4a+2b+c$, right>