# Thread: bilinear form

1. ## 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.

2. You can easily verify that for f:

(1) $f(\cdot,k)$ is linear.
(2) $f(h,\cdot)$ is linear.
(3) $f(h,k)=f(k,h)$ and
(4) $f(th,tk)=t^2f(h,k), \ t\in \mathbb{R}.$

So f is a symmetric bilinear form.

Its matrix with respect to the basis B will have entries $[f]_{ij}=[f(x^i,x^j)]=\left[\int_0^{\infty}x^{i+j}{\rm e}^{-x}dx\right]$, which you can find by repeated use of the integration by parts formula.