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.

Name: Show the details please. Suppose V is a vector space that is not neces
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Description: Show the details please. 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 setT(S) = {wE

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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Show the details please.

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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