Let F be a field and let V, be the F-vector space of polynomials in F[x] of degree
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.3: Factorization In F [x]
Problem 9E: Let be a field. Prove that if is a zero of then is a zero of
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![Let F be a field and let V, be the F-vector space of polynomials in F[x] of degree <n. Which of the
following maps are linear transformations?
(a) The map D : n → Vn-1 sending p(x) to p'(x) = P(x).
(b) The map I : Vn → Vn+1 sending p(x) to P(x) := * p(t)dt.
(c) The map sending p(x) to p(x)².
(d) The map sending p(x) to p(x + 1) – p(x).
(e) The map sending p(x) to p(x)p(x – 1).
Provide a brief explanation -- ideally not more than one or two complete sentences -- for each of your
answers.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe5046824-623a-41fb-a6e0-643b469382f0%2F1a26f890-c582-40f2-a490-2a685a0bca91%2F9v4t0e_processed.png&w=3840&q=75)
Transcribed Image Text:Let F be a field and let V, be the F-vector space of polynomials in F[x] of degree <n. Which of the
following maps are linear transformations?
(a) The map D : n → Vn-1 sending p(x) to p'(x) = P(x).
(b) The map I : Vn → Vn+1 sending p(x) to P(x) := * p(t)dt.
(c) The map sending p(x) to p(x)².
(d) The map sending p(x) to p(x + 1) – p(x).
(e) The map sending p(x) to p(x)p(x – 1).
Provide a brief explanation -- ideally not more than one or two complete sentences -- for each of your
answers.
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