Question

# Let the vector space P^{2} have the inner product langle p,qrangle=int_{-1}^{1} p(x)q(x)dx. Find the following for p = 1 and q = x^{2}. (a) ⟨p,q⟩ (b) ∥p∥ (c) ∥q∥ (d) d(p,q)

Vectors and spaces
Let the vector space $$P^{2}$$
have the inner product $$\langle p,q\rangle=\int_{-1}^{1} p(x)q(x)dx.$$
Find the following for $$p = 1\ and\ q = x^{2}.$$
$$(a) ⟨p,q⟩ (b) ∥p∥ (c) ∥q∥ (d) d(p,q)$$

2021-03-13

Given that the vector space P have the inner product
$$\langle p,q\rangle=\int_{-1}^{1} p(x)q(x)dx.$$
Also given that $$p = 1\ and\ g = x^{2}$$
We calculate the value of  as follows:
$$⟨p, q⟩ =\int_{-1}^{1} p(x)q(x)dx$$
$$= \int_{-1}^{1} x^{2} dx$$
Therefore, the value of $$\langle p,q\rangle=\frac{2}{3}$$