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)
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.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F01b4eff1-1c92-4d86-9563-264f29556131%2Fe73ded3c-5341-48e9-ba61-867e65fae4f6%2Fkuz9gh_processed.jpeg&w=3840&q=75)
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
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