3. Let T: V→ V be a linear operator on a finite-dimensional vector space V. Let B = {₁,..., Un} be an ordered basis for V. Let k be an integer with 0
3. Let T: V→ V be a linear operator on a finite-dimensional vector space V. Let B = {₁,..., Un} be an ordered basis for V. Let k be an integer with 0
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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![3. Let T: V→ V be a linear operator on a finite-dimensional vector space V. Let
B = {₁,..., Un} be an ordered basis for V. Let k be an integer with 0 <k <n,
and let ₁ = {₁,..., Uk}, B₂ = {Uk+1,..., Un}, W₁ = span(3₁), and W₂ = span(3₂).
Show that the following two conditions are equivalent.
(a) The matrix [T] is block diagonal of the form
[T]₁ = ({
0
0 A₂
for some k x k matrix A₁ and some (n - k) × (n − k) matrix A₂.
(b) Both W₁ and W₂ are T-invariant subspaces of V.
Moreover, show that when these conditions are satisfied, we have A₁ = [Tw₁]₁ and
A2 = [Tw₂]8₂.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2e948f6-fd6f-485f-942e-c931230f8579%2Fc5a3bc41-d551-4158-8a08-d1d2a72c546a%2F6hxgggf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. Let T: V→ V be a linear operator on a finite-dimensional vector space V. Let
B = {₁,..., Un} be an ordered basis for V. Let k be an integer with 0 <k <n,
and let ₁ = {₁,..., Uk}, B₂ = {Uk+1,..., Un}, W₁ = span(3₁), and W₂ = span(3₂).
Show that the following two conditions are equivalent.
(a) The matrix [T] is block diagonal of the form
[T]₁ = ({
0
0 A₂
for some k x k matrix A₁ and some (n - k) × (n − k) matrix A₂.
(b) Both W₁ and W₂ are T-invariant subspaces of V.
Moreover, show that when these conditions are satisfied, we have A₁ = [Tw₁]₁ and
A2 = [Tw₂]8₂.
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