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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![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 (-,-).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c741b4f-da7f-48ba-b84e-2097952246cb%2F7cf57b16-77b1-407c-b3ab-ba8123cd4962%2F1s0m7hr_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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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