Showing a particular function on the space of polynomials w/ degree $ \le 2$ is a quadratic form, and computing signature

Question

I've been a long way from linear algebra but I have to go back to it for an exam, and I've found myself stuck on the following question.

Define $Q$ on the space of all polynomials with degree at most 2 by the following:

(apologies for lack of formatting)

$$Q(P(t)) = \int_{-1}^1 (p(t))^2 dt - \int_{-1}^1 (p'(t))^2 dt$$

Show $Q$ is a quadratic form and compute the signature.

I've worked with bilinear forms and such before, but I don't know how to approach this. I'm sure the second part would also be fine if I could get the first. All help would be appreciated, or a push in the right direction.

Thanks.

Answer 1

Hints:

After computing $(Q(p+q)-Q(p)-Q(q))/2$ and determining that the result is a bilinear form, you can take a look at the most obvious basis for this vector space: $\{1,x,x^2\}$.

As luck would have it, $1$ and $x$ are orthogonal, and so are $x$ and $x^2$. The only problem is that $1$ and $x^2$ aren't orthogonal. One application of Gram-Schmidt later, you have an orthogonal basis, and you can count signs to get the signature.