Problem 2 : Let T : R3[r]→ R3[x] be the linear operator defined as T(p(x)) = (x + 6)p (x) + p"(x), where p'(x) and p"(x) denote the first and second derivatives of p(x) respectively. a) Show that T is linear. b) Find a basis for Ker(T) and determine dim(Ker(T)). c) Find a basis for Im(T) and determine dim(Im(T)). d) Is T an isomorphism?
Problem 2 : Let T : R3[r]→ R3[x] be the linear operator defined as T(p(x)) = (x + 6)p (x) + p"(x), where p'(x) and p"(x) denote the first and second derivatives of p(x) respectively. a) Show that T is linear. b) Find a basis for Ker(T) and determine dim(Ker(T)). c) Find a basis for Im(T) and determine dim(Im(T)). d) Is T an isomorphism?
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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please send handwritten solution for part d
![Problem 2
: Let T: R3[x]
R3[x] be the linear operator defined as
T(p(x)) = (x+6)p (x)+ p"(x),
where p'(x) and p"(x) denote the first and second derivatives of p(x) respectively.
a) Show that T is linear.
b) Find a basis for Ker(T) and determine dim(Ker(T)).
c) Find a basis for Im(T) and determine dim(Im(T)).
d) Is T an isomorphism?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F26a77bed-bbca-410f-a4fd-d651a789a370%2F15b9c4e3-9a32-4490-b5be-33c8a01fe85b%2F8t7igam_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 2
: Let T: R3[x]
R3[x] be the linear operator defined as
T(p(x)) = (x+6)p (x)+ p"(x),
where p'(x) and p"(x) denote the first and second derivatives of p(x) respectively.
a) Show that T is linear.
b) Find a basis for Ker(T) and determine dim(Ker(T)).
c) Find a basis for Im(T) and determine dim(Im(T)).
d) Is T an isomorphism?
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