Problem 2 (Linear independence in the polynomial space) (a) Prove that the k polynomials n Pi(x) = Σaijx³ for i = 1, ..., k j=0 are linearly independent if and only if the (n + 1) × k matrix AT has rank k, where A is the k × (n+1) matrix whose entries are the coefficients aij for 1 ≤ i ≤ k, 0 ≤ j ≤ n . (b) Formulate a similar matrix condition for testing whether or not another polynomial q(x) lies in their span. (c) Use (a) to determine whether the polynomials P1, P2, P3, P4, P5 are linearly independent or linearly depen- dent, where P₁(x) = x³- 1, P2(x) = x³ − 2x +4, - Pз(x) = x² - 4x, P4(x) = x² + 1, P5 (x) = −x² + 4x³ +2x+1 (d) Does the polynomial q(x) = x³ lie in their span? If so, find a linear combination that adds up to q(x).
Problem 2 (Linear independence in the polynomial space) (a) Prove that the k polynomials n Pi(x) = Σaijx³ for i = 1, ..., k j=0 are linearly independent if and only if the (n + 1) × k matrix AT has rank k, where A is the k × (n+1) matrix whose entries are the coefficients aij for 1 ≤ i ≤ k, 0 ≤ j ≤ n . (b) Formulate a similar matrix condition for testing whether or not another polynomial q(x) lies in their span. (c) Use (a) to determine whether the polynomials P1, P2, P3, P4, P5 are linearly independent or linearly depen- dent, where P₁(x) = x³- 1, P2(x) = x³ − 2x +4, - Pз(x) = x² - 4x, P4(x) = x² + 1, P5 (x) = −x² + 4x³ +2x+1 (d) Does the polynomial q(x) = x³ lie in their span? If so, find a linear combination that adds up to q(x).
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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