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Let L: V→V be a linear transformation. a) If U₁, ..., Un are L-invariant subspaces, show that their sum is L-invariant. U₁+U₂+ + Un = {u₁ + U₂ + ... + Un : Ui € U₁}

Let L: V→V be a linear transformation. a) If U₁, ..., Un are L-invariant subspaces, show that their sum is L-invariant. U₁+U₂+ + Un = {u₁ + U₂ + ... + Un : Ui € U₁}

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.6: The Matrix Of A Linear Transformation
Problem 43EQ
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4. (*) Let L: V → V be a linear transformation. a) If U₁, ..., Un are L-invariant subspaces, show that their sum U₁+U₂ + + Un = {u₁ + U₂ + ··· + Un • Ui € Ui} is L-invariant. b) Suppose that L is invertible, that U is an L-invariant subspace of V. For any v U, show that L(v) & U. c) If V has dimension n and U is an L-invariant subspace of dimension m, prove that there is a basis B for V so that the matrix of L with respect to B has the following shape: 0 0 a1,1 a1.m 0 0 am, 1 am,m [L] B = b1,1 b1,m €1,1 C1,n-m bn-m,1 bn-m,m Cn-m,1 Cn-m,n-m
Transcribed Image Text:4. (*) Let L: V → V be a linear transformation. a) If U₁, ..., Un are L-invariant subspaces, show that their sum U₁+U₂ + + Un = {u₁ + U₂ + ··· + Un • Ui € Ui} is L-invariant. b) Suppose that L is invertible, that U is an L-invariant subspace of V. For any v U, show that L(v) & U. c) If V has dimension n and U is an L-invariant subspace of dimension m, prove that there is a basis B for V so that the matrix of L with respect to B has the following shape: 0 0 a1,1 a1.m 0 0 am, 1 am,m [L] B = b1,1 b1,m €1,1 C1,n-m bn-m,1 bn-m,m Cn-m,1 Cn-m,n-m
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