Suppose V is a vector space that is not necessarily finite-dimensional. Suppose S is a subset of V that is not necessarily finite. Given a linear transformation T : V → V , define the image of S under T to be the set T(S) = {wE V : 3v E S, w = T(v)} (1) Prove that if T(v) is linearly independent, then S is linearly independent. (2) Justify whether it is true that if S is linearly independent then T(v) must be linearly independent.
Suppose V is a vector space that is not necessarily finite-dimensional. Suppose S is a subset of V that is not necessarily finite. Given a linear transformation T : V → V , define the image of S under T to be the set T(S) = {wE V : 3v E S, w = T(v)} (1) Prove that if T(v) is linearly independent, then S is linearly independent. (2) Justify whether it is true that if S is linearly independent then T(v) must be linearly independent.
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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Transcribed Image Text:Suppose V is a vector space that is not necessarily finite-dimensional.
Suppose S is a subset of V that is not necessarily finite.
Given a linear transformation T : V → V , define the image of S under T to be the set
T(S) = {wE V : 3v E S, w = T(v)}
(1) Prove that if T(v) is linearly independent, then S is linearly independent.
(2) Justify whether it is true that if S is linearly independent then T(v) must be linearly
independent.
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