Complementary polynomial space

[Mindscrape]

Let $$V = P^{(4)}$$ denote the space of quartic polynomials, with the L^2 inner product

$$<p,q>= \int_{-l}^l p(x)q(x)dx$$

Let $$W = P^2$$ be the subspace of quadratic polynomials.
a) Write down the conditions that a polynomial $$p \in P^{(4)}$$ must satisfy in order to belong to the orthogonal complement Wperp.
b) Find a basis for and the dimension of Wperp.
c) Find an orthogonal basis for Wperp.

The first part just goes off the definition of a complementary subspace.
$$W_{perp} = (q(x) = a + bx + cx^2 + dx^3 + ex^4 | <p,1>=<p,x>=<p,x^2>=0)$$

It looks like the second part wants me to actually do the calculation, but that looks like a lot of work to multiply it out. Is that really what it is asking?

The last part should be easy, because I can just take the basis from part b) and apply the Gram-Schmidt method.

 

[Dick]

What is 'a lot of work to multiply out'? It's just monomials times a polynomial integrated over [-1,1]. You get three conditions on a,b,c,d,e. It doesn't get too much more straightforward than that.