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}: %3D 2 A = |0 3 2 0 [1 0. ēj = e2 = 1 A. Find the eigenvalues {A;} for A. E Find the set of orthornormal eigenvectors {pi}. E Show that the P = [P1 P2 p3] is symmetric, i.e. PT = P-1 9. Show that D = PT AP = 0. 0. 0. 0. Bonus. Show that pi" Api =d1 A 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]B = P [Pi]B-

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Please help me the first 3 parts. Part A part B and part C. Please make sure type by computer although 

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
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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