
bilinear form
Hi,
Can anyone help with the following question:
Let Vn be the vector space of of polynomials,
h element of R[x],
deg(f)<= n.
For h,k element of Vn  define:
f(h,k) = integral between 0 and infinity of [ h(x)k(x)(e^x)] dx
a) Show that f is symmetric bilinear form
b) Let B be the basis {1,X, . . . X^n} of Vn. Find [f] w.r.t B.

You can easily verify that for f:
(1) $\displaystyle f(\cdot,k)$ is linear.
(2) $\displaystyle f(h,\cdot)$ is linear.
(3) $\displaystyle f(h,k)=f(k,h)$ and
(4) $\displaystyle f(th,tk)=t^2f(h,k), \ t\in \mathbb{R}.$
So f is a symmetric bilinear form.
Its matrix with respect to the basis B will have entries $\displaystyle [f]_{ij}=[f(x^i,x^j)]=\left[\int_0^{\infty}x^{i+j}{\rm e}^{x}dx\right]$, which you can find by repeated use of the integration by parts formula.