1 The linear transformation T: R → R°, T(F) = Aï, is dehned by the basis B={e1, e2, e3}: [1 0. 27 e2 = ez = 0. A = |0 2 0 %3D 1 A. Find the eigenvalues {A;} for A. E Find the set of orthornormal eigenvectors {pi}. E Show that the P= [Pi P2 P3] is symmetric, i.e. PT = P-1 0. 9. Show that D = PT AP = 0. 0. 0 13]
1 The linear transformation T: R → R°, T(F) = Aï, is dehned by the basis B={e1, e2, e3}: [1 0. 27 e2 = ez = 0. A = |0 2 0 %3D 1 A. Find the eigenvalues {A;} for A. E Find the set of orthornormal eigenvectors {pi}. E Show that the P= [Pi P2 P3] is symmetric, i.e. PT = P-1 0. 9. Show that D = PT AP = 0. 0. 0 13]
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 need the first 4 parts part A part B part c and part D
please makes sure type by computer thanks
![1 The linear transformation T: R3 – R³, T(7) = A, is defined by the matrix A relative to the standard
basis B={e1, e2, ē3}:
%3D
1
0.
0 2
A = 0 30
2 0
ei =
0.
1
0.
1
0.
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
%3D
0.
0.
9. Show that D = PT AP =
%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 = P Pi]B:
%3D](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba35cc5f-959d-4f80-acdd-e9cc0bd31dfa%2F3af7db9b-81dd-4ef4-9b89-34b31ec549bd%2Fek1zic8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1 The linear transformation T: R3 – R³, T(7) = A, is defined by the matrix A relative to the standard
basis B={e1, e2, ē3}:
%3D
1
0.
0 2
A = 0 30
2 0
ei =
0.
1
0.
1
0.
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
%3D
0.
0.
9. Show that D = PT AP =
%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 = P Pi]B:
%3D
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