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2. Let a₁,..., ar be distinct real numbers. For p, q € Pn (R) let (,) by defined by p(ai)q(ai). (a) Prove that (p, p) ≥ 0 for all p. (p, q) = r ΣP i=1 (b) Prove that if r > n, then (,) is positive definite. (c) Prove or disprove: if r ≤ n, then (,) is positive definite.

2. Let a₁,..., ar be distinct real numbers. For p, q € Pn (R) let (,) by defined by p(ai)q(ai). (a) Prove that (p, p) ≥ 0 for all p. (p, q) = r ΣP i=1 (b) Prove that if r > n, then (,) is positive definite. (c) Prove or disprove: if r ≤ n, then (,) is positive definite.

Advanced Engineering Mathematics
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 a₁,..., ar be distinct real numbers. For p, q € Pn (R) let (,) by defined by (p, q) =p(a)q(ai). (a) Prove that (p, p) ≥0 for all p. r i=1 (b) Prove that if r > n, then (,) is positive definite. (c) Prove or disprove: if r ≤n, then (,) is positive definite.
Transcribed Image Text:2. Let a₁,..., ar be distinct real numbers. For p, q € Pn (R) let (,) by defined by (p, q) =p(a)q(ai). (a) Prove that (p, p) ≥0 for all p. r i=1 (b) Prove that if r > n, then (,) is positive definite. (c) Prove or disprove: if r ≤n, then (,) is positive definite.
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