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-
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba35cc5f-959d-4f80-acdd-e9cc0bd31dfa%2Ffc9cba6e-509b-4af8-b443-a0c444c72e58%2Fbf01pjd_processed.jpeg&w=3840&q=75)
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