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(f) What is span(q, r)? (g) What is span(q, r, s)?

(f) What is span(q, r)? (g) What is span(q, r, s)?

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.4: Linear Transformations
Problem 24EQ
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part F G 

Consider the following polynomials as vectors from the vector space P3 (R) (Definition 2.13 in the Sheldon's book): p(z) = 2³-2z+1 q(z) = 2²-1 r(z) = 22³-2 = 2²-22+1 t(z) 2³+az²+b, = z € R. (When asked about span of a list of vectors, you are expected to express it in the following form: span(p, q) = {a32³ + a22²-2a3z + (az-a₂) € P3(R) : a3, a2 € R}.) (a) Are the vectors p, q, r linearly independent? (b) Are the vectors p, q, r, s linearly independent? (c) * For what values of a R and b ER the vectors p, t linearly independent? (d) For what values of a R and b E R the vectors p, q, t linearly independent? (e) For what values of a € R and b ER the vectors p, q, r, s, t linearly independent? (f) What is span(q, r)? What is span(q, r, s)?
Transcribed Image Text:Consider the following polynomials as vectors from the vector space P3 (R) (Definition 2.13 in the Sheldon's book): p(z) = 2³-2z+1 q(z) = 2²-1 r(z) = 22³-2 = 2²-22+1 t(z) 2³+az²+b, = z € R. (When asked about span of a list of vectors, you are expected to express it in the following form: span(p, q) = {a32³ + a22²-2a3z + (az-a₂) € P3(R) : a3, a2 € R}.) (a) Are the vectors p, q, r linearly independent? (b) Are the vectors p, q, r, s linearly independent? (c) * For what values of a R and b ER the vectors p, t linearly independent? (d) For what values of a R and b E R the vectors p, q, t linearly independent? (e) For what values of a € R and b ER the vectors p, q, r, s, t linearly independent? (f) What is span(q, r)? What is span(q, r, s)?
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