(1 point) LetDefine the linear transformation T: R³ → R¹ by T(z) = Az. Find a vector whose image under T is b.1=Is the vector #unique? chooseA =63-3-36 7-2 -3-6 4-5-4-37and 6-2822062

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Answered: Define the linear transformation T: R³…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Linear Transformations

Section6.2: The Kernewl And Range Of A Linear Transformation

Problem 68E

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Guided Proof Let B be an invertible nn matrix. Prove that the linear transformation T:Mn,nMn,n represented by T(A)=AB is an isomorphism. Getting started: To show that the linear transformation is an isomorphism, you need to show that T is both onto and one-to-one. (i) T is a linear transformation with vector spaces of equal dimension, so by Theorem 6.8, you only need to show that T is one-to-one. (ii) To show that T is one-to-one, you need to determine the kernel of T and show that it is {0} Theorem 6.6. Use the fact that B is an invertible nn matrix and that T(A)=AB. (iii) Conclude that T is an isomorphism.
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Define the linear transformation T: R³ → R¹ by T() = A. Find a vector whose image under T is b. 2= Is the vector a unique? choose