Let U, V, W be finite dimensional vector spaces over F, and S : U → V,T:V → W linear transformations. (c) Prove that if T is injective, then N(S) = N(T o S). (d) Is the converse of (c) true? (Concretely: Is it true that if N(S) = N(T o S) then T is injective?) If true, prove it. If false, give a counter example.

Let U, V, W be finite dimensional vector spaces over F, and S : U → V,T:V → W linear transformations. (c) Prove that if T is injective, then N(S) = N(T o S). (d) Is the converse of (c) true? (Concretely: Is it true that if N(S) = N(T o S) then T is injective?) If true, prove it. If false, give a counter example.

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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