Problem Solving - Linear Transformation
Problem Solving - Linear Transformation M.Velrajan We continue our discussion on Problem Solving - Pure Mathematics with elementary exercises on linear transformations given in Topics in Algebra, by I. N. Herstein, second edition. Let V be a finite dimensional vector space over a field F and A(V) = Hom (V, V) be the algebra of all linear transformations of V into V. 1. S ∈ A(V) is regular if and only if whenever v 1 , . . . , v n ∈ V are linearly independent, then v 1 S, . . . , v n S are also linearly independent. Suppose S ∈ A(V) is regular. Let v 1 , . . . , v n ∈ V be linearly independent. To prove v 1 S, . . . , v n S are linearly independent we prove 𝛂 1 v 1 S + . . . + 𝛂 n v n S = 0 ⟹ 𝛂 1 = . . . = 𝛂 n = 0. Then 𝛂 1 v 1 S + . . . + 𝛂 n v n S = 0 ⟹ (𝛂 1 v 1 + . . . + 𝛂 n v n ) S = 0 (since S is linear) ...