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2. Let P2 be the vector space of polynomials of degree at most 2. Let T: P2 → P₂ by T (p(x)) = p(x) + x p(x). a) Show that T is a linear transformation b) Recall T: P₂ → P2 is defined by T (p(x)) = p(x) + x p(x). Detemine the kernel of T. c) Recall T: P₂ → P2 is defined by T (p(x)) = p(x) + x p(x). Show that T is an isomorphism.

2. Let P2 be the vector space of polynomials of degree at most 2. Let T: P2 → P₂ by T (p(x)) = p(x) + x p(x). a) Show that T is a linear transformation b) Recall T: P₂ → P2 is defined by T (p(x)) = p(x) + x p(x). Detemine the kernel of T. c) Recall T: P₂ → P2 is defined by T (p(x)) = p(x) + x p(x). Show that T is an isomorphism.

Advanced Engineering Mathematics
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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2. Let P2 be the vector space of polynomials of degree at most 2. Let T: P2 → P2 by T (p(x)) = p(x) + x p(x). a) Show that T is a linear transformation b) Recall T: P₂ → P₂ is defined by T (p(x)) = p(x) + x dp(x). Detemine the kernel of T. c) Recall T: P₂ → P₂ is defined by T (p(x)) = p(x) + x p(x). Show that T is an isomorphism. d) Determine T−¹.
Transcribed Image Text:2. Let P2 be the vector space of polynomials of degree at most 2. Let T: P2 → P2 by T (p(x)) = p(x) + x p(x). a) Show that T is a linear transformation b) Recall T: P₂ → P₂ is defined by T (p(x)) = p(x) + x dp(x). Detemine the kernel of T. c) Recall T: P₂ → P₂ is defined by T (p(x)) = p(x) + x p(x). Show that T is an isomorphism. d) Determine T−¹.
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