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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.)

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.)

Elementary Linear Algebra (MindTap Course List)
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.)
Transcribed Image Text: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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