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A permutation operator, P, in a 3 dimensional linear vector space with basis vectors ê, ê, ê, has the following operation on the basis vectors: Pê₁ = ê3 Pê₂ = ê¹ Pê3 = ê₂. a) Derive P, the matrix form of the operator P, with the same basis used for the input and output spaces. Note: If you cannot solve this part then use the following matrix for the next two parts: Го о 17 100 10 b) Show that P3 = I, where I is the identity matrix. c) Hence show that for positive integers n and m: I for n P"=P P² for n = 3m = 3m +1 for n = 3m +2

A permutation operator, P, in a 3 dimensional linear vector space with basis vectors ê, ê, ê, has the following operation on the basis vectors: Pê₁ = ê3 Pê₂ = ê¹ Pê3 = ê₂. a) Derive P, the matrix form of the operator P, with the same basis used for the input and output spaces. Note: If you cannot solve this part then use the following matrix for the next two parts: Го о 17 100 10 b) Show that P3 = I, where I is the identity matrix. c) Hence show that for positive integers n and m: I for n P"=P P² for n = 3m = 3m +1 for n = 3m +2

Advanced Engineering Mathematics
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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A permutation operator, P, in a 3 dimensional linear vector space with basis vectors ê, ê, ê, has the following operation on the basis vectors: Pê₁ = ê3 Pê₂ = ê¹ Pê3 = ê₂. a) Derive P, the matrix form of the operator P, with the same basis used for the input and output spaces. Note: If you cannot solve this part then use the following matrix for the next two parts: Го о 17 100 10 b) Show that P3 = I, where I is the identity matrix. c) Hence show that for positive integers n and m: I for n P"=P P² for n = 3m = 3m +1 for n = 3m +2
Transcribed Image Text:A permutation operator, P, in a 3 dimensional linear vector space with basis vectors ê, ê, ê, has the following operation on the basis vectors: Pê₁ = ê3 Pê₂ = ê¹ Pê3 = ê₂. a) Derive P, the matrix form of the operator P, with the same basis used for the input and output spaces. Note: If you cannot solve this part then use the following matrix for the next two parts: Го о 17 100 10 b) Show that P3 = I, where I is the identity matrix. c) Hence show that for positive integers n and m: I for n P"=P P² for n = 3m = 3m +1 for n = 3m +2
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