16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 8.1.11

Inner Product Spaces

Mathematics — Linear Algebra · inner-products-polynomials

Problem

Show that the formula

\left(\sum_j a_j x^j \bigg| \sum_k b_k x^k\right) = \sum_{j,k} \frac{a_j b_k}{j + k + 1}

defines an inner product on the space

R[x]

of polynomials over the field

R

. Let

W

be the subspace of polynomials of degree less than or equal to

n

. Restrict the above inner product to

W

, and find the matrix of this inner product on

W

, relative to the ordered basis

\{1, x, x^2, \ldots, x^n\}

. (Hint: To show that the formula defines an inner product, observe that

(f|g) = \int_0^1 f(t)g(t)\, dt

and work with the integral.)