2. Let (,): P3 × P3 → R defined by(p,q) = p(0)q(0) + 2p'(0)q'(0) + 3p"(0)q"(0) + p''(0)q''(0)for every p, q E P3.(a) Prove that (P3,(',;)) is an inner product space.(b) Let S = {p1, p2, P3} C P3, where p1(x) = 1 – x², P2(x) = 3 and%3D%3DP3(x) = x³. Given that S is linearly independent, use the Gram-Schmidt Orthogonalization Process to find an orthonormal basisfor span(S) with respect to the inner product (-,-).

Given, ,:ℙ3×ℙ3→ℝ defined by, p,q=p(0)q(0)+2p'(0)q'(0)+3p''(0)q''(0)+p'''(0)q'''(0) ∀p,q∈ℙ3 a) To…

2. Let (,): P3 × P3 → R defined by (p, q) = p(0)q(0) + 2p'(0)q'(0) + 3p"(0)q"(0) + p''(0)q"'(0) for every p,q E P3. (a) Prove that (P3, (,;)) is an inner product space. (b) Let S = {p1, P2, P3} c P3, where p,(x) = 1 – x², P2(x) = 3 and P3(x) = x³. Given that S is linearly independent, use the Gram -Schmidt Orthogonalization Process to find an orthonormal basis for span(S) with respect to the inner product (,-).

2. Let (,): P3 × P3 → R defined by (p, q) = p(0)q(0) + 2p'(0)q'(0) + 3p"(0)q"(0) + p''(0)q"'(0) for every p,q E P3. (a) Prove that (P3, (,;)) is an inner product space. (b) Let S = {p1, P2, P3} c P3, where p,(x) = 1 – x², P2(x) = 3 and P3(x) = x³. Given that S is linearly independent, use the Gram -Schmidt Orthogonalization Process to find an orthonormal basis for span(S) with respect to the inner product (,-).

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

2. Let (,): P3 × P3 → R defined by
(p,q) = p(0)q(0) + 2p'(0)q'(0) + 3p"(0)q"(0) + p''(0)q''(0)
for every p, q E P3.
(a) Prove that (P3,(',;)) is an inner product space.
(b) Let S = {p1, p2, P3} C P3, where p1(x) = 1 – x², P2(x) = 3 and
%3D
%3D
P3(x) = x³. Given that S is linearly independent, use the Gram
-Schmidt Orthogonalization Process to find an orthonormal basis
for span(S) with respect to the inner product (-,-).

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