Let {v1, v2, . . . , vn} be a basis for a vector space V. Prove that if a linear transformation T: V→V satisfies T(vi) = 0 for i = 1, 2, . . . , n, then T is the zero transformation.Getting Started: To prove that T is the zero transformation, you need to show that T(v) = 0 for every vector v in V.(i) Let v be an arbitrary vector in V such that v = c1v1 + c2v2 + . . . + cnvn.(ii) Use the definition and properties of linear transformations to rewrite T(v) as a linear combination of T(vi).(iii) Use the fact that T(vi) = 0 to conclude that T(v) = 0, making T the zero transformation.

Let {v1, v2, . . . , vn} be a basis for a vector space V. Prove that if a linear transformation T: V→V satisfies T(vi) = 0 for i = 1, 2, . . . , n, then T is the zero transformation.Getting Started: To prove that T is the zero transformation, you need to show that T(v) = 0 for every vector v in V.(i) Let v be an arbitrary vector in V such that v = c1v1 + c2v2 + . . . + cnvn.(ii) Use the definition and properties of linear transformations to rewrite T(v) as a linear combination of T(vi).(iii) Use the fact that T(vi) = 0 to conclude that T(v) = 0, making T the zero transformation.

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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