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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.

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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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