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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?

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter6: Linear Transformations
Section6.CR: Review Exercises
Problem 65CR
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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.
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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