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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).

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.2: Direct Methods For Solving Linear Systems
Problem 3CEXP
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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).
Transcribed Image Text: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).
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