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 : V → Vn-1 sending p(x) to p' (x) = p(x). (b) The map I : V, → Vn+1 sending p(x) to P(x) := f* p(1)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.
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 : V → Vn-1 sending p(x) to p' (x) = p(x). (b) The map I : V, → Vn+1 sending p(x) to P(x) := f* p(1)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.
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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Question
![Q2
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 : V, → Vn=1 sending p(x) to p' (x) = p(x).
(b) The map I : Vn
→ Vn+1 sending p(x) to P(x) := f 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%2Fb90e355c-abe9-45d7-b8d1-631ace6b1e69%2Ffaafb713-4e83-4dd5-9d73-bc1deff68813%2F70pyut9_processed.png&w=3840&q=75)
Transcribed Image Text:Q2
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 : V, → Vn=1 sending p(x) to p' (x) = p(x).
(b) The map I : Vn
→ Vn+1 sending p(x) to P(x) := f 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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