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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

Advanced Engineering Mathematics
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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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
Expert Solution
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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