# Finding a Subspace for a 2-Degree Polynomial Through Integration

I'm working on solving a problem that concerns polynomials, vectors, and inner products. A link to a photo of exact question has been posted below.

Basically, it goes as follows: P2 is a vector space of all polynomials on the interval [-1, 1] with a degree ≤ 2 (so anything less than x to the power 2). The given vectors are p = p(x) and q = q(x).

I am supposed to define the inner product <p, q> on the integral ∫p(x)q(x)dx from -1 to 1.

Furthermore, W = span{x}, and is a subspace of all polynomials of the form λx (i.e., generated by x).

Following this, the question also wants me to A) find the subspace W⊥, and B) to verify that WW⊥ is equal to zero.

I know how to verify or prove the last part (part B)), but I'm not quite sure how to define the inner product or how to find the orthogonal complement itself. I know that, when given a series of equations or a matrix, all you have to do is put the matrix is row echelon form to find a basis for the orthogonal complement. All of the similar problems in my notes and in the textbook have given values for p and q. I tried making p = x and q = x^2, which proved that they were orthogonal to each other upon evaluating the integral of their product.

If anyone can give me some insight on how to approach this problem, I'd be most appreciative. Thank you in advance!

Since you can compute everything explicitly, it suffices to define $$p(x)=x\quad\text{and}\quad q(x)=ax^2+bx+c,$$ and compute explicitly $$\langle p,q\rangle=\int_{-1}^1x(ax^2+bx+c)\,dx=\frac{2b}3.$$ This implies that $$W^\bot=\{q:\langle p,q\rangle=0\}=\{ax^2+c:a,c\in\mathbb R\}.$$