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
Vectors and spaces
asked 2021-03-12
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 \(p = 1\ and\ q = x^{2}.\)
\((a) ⟨p,q⟩ (b) ∥p∥ (c) ∥q∥ (d) d(p,q)\)

Answers (1)

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 \(⟨p, q⟩ = \frac{2}{3}
0

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