Question

Let the vector space P2 have the inner product

⟨p,q⟩=∫4−4p(x)q(x)dx

Find the following for p=2x, q=5x2.

Solution

(i) ⟨p,q⟩=

(ii) d(p,q)=

(iii) ∥p∥=

(iv) ∥q∥=

Answer 1

Given the inner product:

(p, q) = √4 - 4∫p(x)q(x)dx

And the vectors p = 2x and q = 5x^2, we can calculate their inner product as follows:

(p, q) = √4 - 4∫2x * 5x^2 dx = √4 - 4 * (5/3) * x^3 | from 0 to 1
(p, q) = √4 - 4 * (5/3) * 1^3 = √4 - 20/3

Therefore, the inner product of p = 2x and q = 5x^2 is √4 - 20/3.