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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