Verify that the following is a linear transformation: T: C([0, 1]) → C([0, 1]) T(ƒ) = [ x²ƒ(x) dx where the constant of integration is always zero. Would this function be a linear transforma- tion if the constant of integration was one instead?
Verify that the following is a linear transformation: T: C([0, 1]) → C([0, 1]) T(ƒ) = [ x²ƒ(x) dx where the constant of integration is always zero. Would this function be a linear transforma- tion if the constant of integration was one instead?
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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![Verify that the following is a linear transformation:
T: C([0, 1])→ C([0, 1])
T(f) =/z³f(x)dr
[x²
where the constant of integration is always zero. Would this function be a linear transforma-
tion if the constant of integration was one instead?
Note: This linear transformation has ker(T) = {0}, however, ran(T) consists solely of func-
tions which are differentiable. Not all continuous functions on [0, 1] are differentiable, so this
is an example of a linear transformation, from a vector space to itself, with trivial kernel that
is not surjective. This can only happen with infinite dimensional vector spaces thanks to the
Rank-Nullity Theorem, and it is one of the pathological things about infinite dimensional vector
spaces.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4217e83c-b4b2-47da-9d7a-2d6031448fca%2Fe24128d4-6d21-41b0-a3f5-90646540d549%2Fbao1lng_processed.png&w=3840&q=75)
Transcribed Image Text:Verify that the following is a linear transformation:
T: C([0, 1])→ C([0, 1])
T(f) =/z³f(x)dr
[x²
where the constant of integration is always zero. Would this function be a linear transforma-
tion if the constant of integration was one instead?
Note: This linear transformation has ker(T) = {0}, however, ran(T) consists solely of func-
tions which are differentiable. Not all continuous functions on [0, 1] are differentiable, so this
is an example of a linear transformation, from a vector space to itself, with trivial kernel that
is not surjective. This can only happen with infinite dimensional vector spaces thanks to the
Rank-Nullity Theorem, and it is one of the pathological things about infinite dimensional vector
spaces.
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