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}. (*) Let L: V→ V be a linear transformation. a) If U₁,..., Un are L-invariant subspaces, show that their sum U₁+U₂ + + Un = {₁+₂+ + unu₂ € 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,m 0 0 [L]B a1,1 am,1 b₁,1 ⠀ ba-m,1 Gm,m b1.m bn-m,m 0 C1. Cn-m,1 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}. (*) Let L: V→ V be a linear transformation. a) If U₁,..., Un are L-invariant subspaces, show that their sum U₁+U₂ + + Un = {₁+₂+ + unu₂ € 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,m 0 0 [L]B a1,1 am,1 b₁,1 ⠀ ba-m,1 Gm,m b1.m bn-m,m 0 C1. Cn-m,1 0 C1,n-m Cn-m,n-m
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please answer a) , b) and c)
![Definition:
Let L: VV 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}.
(*) Let L: V→ V be a linear transformation.
a) If U₁,..., Un are L-invariant subspaces, show that their sum
U₁+U₂ + ¹ + Un = {U₁ + U₂ + ··· + Un : 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:
0
a1,1
[L]B =
am,1
b₁,1
bn-m.1
⠀⠀
a1,m
⠀
am.m
b1.m
bn-m.m
0
E
0
€1,1
Cn-m,1
0
C1,n-m
Cn-m,n-m](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc43bd3e6-5b65-4fb3-8d46-54613faca65d%2F504809fa-706a-4f6f-be9c-06a42b3200c3%2Fwtdtlhq_processed.png&w=3840&q=75)
Transcribed Image Text:Definition:
Let L: VV 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}.
(*) Let L: V→ V be a linear transformation.
a) If U₁,..., Un are L-invariant subspaces, show that their sum
U₁+U₂ + ¹ + Un = {U₁ + U₂ + ··· + Un : 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:
0
a1,1
[L]B =
am,1
b₁,1
bn-m.1
⠀⠀
a1,m
⠀
am.m
b1.m
bn-m.m
0
E
0
€1,1
Cn-m,1
0
C1,n-m
Cn-m,n-m
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