Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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.
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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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