Let V Ple 2 be equipped with the inner product p q integra

Let V = P_le 2 be equipped with the inner product (p, q) = integral^1_-1 p(x) q(x)/squareroot 1-x^2 dx. Prove that the set p_1(x) = 1, p_2(x) = x, p_3(x) = 2x^2 - 1 is an orthogonal basis in V.

Solution

For p1(x)

put px)=1 in the integration.

In particular, if the inner product of two nonzero vectors is zero, then the angle between those two vectors is /2 radians, and therefore it is natural to call such vectors orthogonal (perpendicular).

when you will put p1(x) in the integration then u calculate integration as a function of q(x) it will become tan^-1 q(X)

i.e. 45-45= 0 hence both p(X) and q(X) are orthogonal in nature.

similarly put p2(x) and p3(x) solve the integration u will get the answer zero which shows <p,q> are orthogonal in nature.