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4) Prove that there does not exist a linear map T: R - R such that range T null T (or equivalently range T = ker T'). %3D

4) Prove that there does not exist a linear map T: R - R such that range T null T (or equivalently range T = ker T'). %3D

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.CR: Review Exercises
Problem 47CR: Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}
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equivalently range T = ker T) 4) Prove that there does not exist a linear map T: RS → R$ such that range T = null T (or equivalently range T = ker T). 5) Suppose DE L(R[x], R2[x]) is the differentiation map defined by Dp = p'. Recall F,[x] is the vector space of all polynomials over a field F of degree at most n and has a u) Find a haaia m Lul
Transcribed Image Text:equivalently range T = ker T) 4) Prove that there does not exist a linear map T: RS → R$ such that range T = null T (or equivalently range T = ker T). 5) Suppose DE L(R[x], R2[x]) is the differentiation map defined by Dp = p'. Recall F,[x] is the vector space of all polynomials over a field F of degree at most n and has a u) Find a haaia m Lul
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Step 1

4 We have to prove that there does not exist a linear map T:R5→R5such that rangeT=nullT.

We know that the range-nullity theorem is given by

RankT+nullityT=dimV, where Vis the vector space.

Here, V=R5

dimV=5

Let us suppose rangeT=nullT=m

⇒m+m=5⇒2m=5⇒m=52⇒m=2.5

Which is not possible.

Hence, it contradict the statement rangeT=nullT=m.

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