16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 8.2.9

Inner Product Spaces

Mathematics — Linear Algebra · orthogonal-complement-gram-schmidt

Problem

Let

V

be the subspace of

R[x]

of polynomials of degree at most 3. Equip

V

with the inner product

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

(a) Find the orthogonal complement of the subspace of scalar polynomials. (b) Apply the Gram-Schmidt process to the basis

\{1, x, x^2, x^3\}