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)

Question

Let the vector space

P^{2}

have the inner product

⟨ p, q ⟩ = \int_{- 1}^{1} p (x) q (x) d x .

Find the following for

p = 1 a n d q = x^{2} .

(a) ⟨ p, q ⟩ (b) ∥ p ∥ (c) ∥ q ∥ (d) d (p, q)

Answer 1

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

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)

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