1 The linear transformation T': K basis B={ei, e2, ē3}: 0. [o 0. [1 0 2 0 3 0 ei = e2 = 0. %3D %3D A = %3D %3D 1 2 0 1 nuolues )-} for A.
1 The linear transformation T': K basis B={ei, e2, ē3}: 0. [o 0. [1 0 2 0 3 0 ei = e2 = 0. %3D %3D A = %3D %3D 1 2 0 1 nuolues )-} for A.
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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Please please address all the question thank you
![1 The linear transformation T : R³ → R³, T(7) = Aï, is defined by the matrix A relative to the standard
basis B={e1, e2, ē3}:
1
0 2
0.
ei =
0.
A = |0 30
1
0.
%3D
0.
1
A. Find the eigenvalues {A;} for A.
2 Find the set of orthornormal eigenvectors {pi}.
E Show that the P = Pi P2 p3] is symmetric, i.e. PT = P-1
%3D
0.
0.
9. Show that D = PT AP =
0.
%3D
0.
T.
E Bonus. Show that pi' Api = A1
4 D is the matrix for T with respect to the eigenbasis B'={p1,p2, P3}.
i. Is P from 3. the transition matrix from B' to B?
Hint: Does [B' B] → |I P'| produce the same P?
ii. What do each of the eigenvectors look like with respect to the eigenbasis?
Hint: Pi]p = PPi]B:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba35cc5f-959d-4f80-acdd-e9cc0bd31dfa%2Fc0ba9f1d-60b4-414d-b91f-0288524cb688%2F5xw8igl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1 The linear transformation T : R³ → R³, T(7) = Aï, is defined by the matrix A relative to the standard
basis B={e1, e2, ē3}:
1
0 2
0.
ei =
0.
A = |0 30
1
0.
%3D
0.
1
A. Find the eigenvalues {A;} for A.
2 Find the set of orthornormal eigenvectors {pi}.
E Show that the P = Pi P2 p3] is symmetric, i.e. PT = P-1
%3D
0.
0.
9. Show that D = PT AP =
0.
%3D
0.
T.
E Bonus. Show that pi' Api = A1
4 D is the matrix for T with respect to the eigenbasis B'={p1,p2, P3}.
i. Is P from 3. the transition matrix from B' to B?
Hint: Does [B' B] → |I P'| produce the same P?
ii. What do each of the eigenvectors look like with respect to the eigenbasis?
Hint: Pi]p = PPi]B:
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