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Definition: Let L: V → V be a linear transformation of a vector space V. A subspace U of V is said to be L-invariant if L(U) CU. Here L(U) = {L(u) | u € U}. 1 (*) 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 + ··· + U₂ · U₂ € U₁} 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: a1,1 41,m 0 E E ⠀ Gm,1 Gm,m 0 [L]B b₁,1 b₁.m €1,1 ⠀ ⠀ bn-m,1 Cn-m,1 -m,m ... 0 C1,n-m Cn-m,n-m

Definition: Let L: V → V be a linear transformation of a vector space V. A subspace U of V is said to be L-invariant if L(U) CU. Here L(U) = {L(u) | u € U}. 1 (*) 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 + ··· + U₂ · U₂ € U₁} 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: a1,1 41,m 0 E E ⠀ Gm,1 Gm,m 0 [L]B b₁,1 b₁.m €1,1 ⠀ ⠀ bn-m,1 Cn-m,1 -m,m ... 0 C1,n-m Cn-m,n-m

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Definition: Let L: V→ V be a linear transformation of a vector space V. A subspace U of V is said to be L-invariant if L(U)CU. Here L(U) = {L(u) | u € U}. 1 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 : U¡ € U₁} 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 @1.1 a1.m ⠀ 0 am,1 am,m 0 C1,n-m [L] B = b1.1 b1.m €1,1 ⠀ bn-m,1 bn-m,m Cn-m,1 Cn-m,n-m
Transcribed Image Text:Definition: Let L: V→ V be a linear transformation of a vector space V. A subspace U of V is said to be L-invariant if L(U)CU. Here L(U) = {L(u) | u € U}. 1 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 : U¡ € U₁} 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 @1.1 a1.m ⠀ 0 am,1 am,m 0 C1,n-m [L] B = b1.1 b1.m €1,1 ⠀ bn-m,1 bn-m,m Cn-m,1 Cn-m,n-m
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