4. I, Let L: V W be a linear transformation from the vector space V to the vector space W such that kerL = {0}. Suppose the vectors 01, 02, ... , Un in V are linearly independent. Is it true that the vectors L(71), L(2),..., L(Tn) in W must be linearly independent? If it is true, prove it. If it is false, provide an example showing why it is false.

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.4: Linear Transformations
Problem 24EQ
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4.
I, Let L: V W be a linear transformation from the vector space V to
the vector space W such that kerL = {0}. Suppose the vectors 01, 02, ... , Un in V are
linearly independent. Is it true that the vectors L(71), L(2),..., L(Tn) in W must be
linearly independent? If it is true, prove it. If it is false, provide an example showing
why it is false.
Transcribed Image Text:4. I, Let L: V W be a linear transformation from the vector space V to the vector space W such that kerL = {0}. Suppose the vectors 01, 02, ... , Un in V are linearly independent. Is it true that the vectors L(71), L(2),..., L(Tn) in W must be linearly independent? If it is true, prove it. If it is false, provide an example showing why it is false.
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