True or False. If the statement is true, prove it. Otherwise, give a counterexample. 1. Let A, B ∈ M_n(F). If nullity(A) = nullity(B) then A and B are similar. 2. The function L : M_n(F) → M_n(F) defined by L(A) = A − A^T is a linear transformation. 3. All nonzero linear transformations T : F^3 → F are onto.
True or False. If the statement is true, prove it. Otherwise, give a counterexample. 1. Let A, B ∈ M_n(F). If nullity(A) = nullity(B) then A and B are similar. 2. The function L : M_n(F) → M_n(F) defined by L(A) = A − A^T is a linear transformation. 3. All nonzero linear transformations T : F^3 → F are onto.
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter6: Linear Transformations
Section6.1: Introduction To Linear Transformations
Problem 74E
Related questions
Question
True or False. If the statement is true, prove it. Otherwise, give a counterexample.
1. Let A, B ∈ M_n(F). If nullity(A) = nullity(B) then A and B are similar.
2. The function L : M_n(F) → M_n(F) defined by L(A) = A − A^T
is a linear transformation.
3. All nonzero linear transformations T : F^3 → F are onto.
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