Let T : R³ → R2 be defined by T(x, y, z) = (x − y, x — 2z).a. Show that T is linear.b. Find ker T and a basis for it.c. Find range T. (Hint: the Rank-Nullity Theorem will help you.)

Let us consider a Transformation T:V→W T is linear if T(v+w)=T(v)+T(w). KerT=v∈V|T(v)=0.…

Answered: Let T: R³ R2 be defined by T(x, y, z) =…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.CM: Cumulative Review

Problem 16CM

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Let T : R³ → R2 be defined by T(x, y, z) = (x − y, x — 2z).
a. Show that T is linear.
b. Find ker T and a basis for it.
c. Find range T. (Hint: the Rank-Nullity Theorem will help you.)

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For part b) do we need to check whether the list (x, x, x/2) is also linearly independent? Since a basis of V is a list of vectors in V that is linearly independent and spans V and we know that span = (x, x, x/2) there is also the requirement of linear independence?

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