prelim2_09Sol - Prelim 2 Solutions Math 2940 November 19,...

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Prelim 2 Solutions Math 2940 November 19, 2009 1. Let A = 0 - 2 2 - 2 0 2 - 2 - 2 4 . (a) (10 points) Determine the eigenvalues of A , and a basis for R 3 consisting of eigenvectors of A . Solution : f A ( λ ) = det ( A - λI 3 ) = - λ 3 + 4 λ 2 - 4 λ = - λ ( λ - 2) 2 = 0 -→ λ = 0 , 2 , 2 λ = 0 : E 0 = ker 0 - 2 2 - 2 0 2 - 2 - 2 4 = span 1 1 1 λ = 2 : E 2 = ker - 2 - 2 2 - 2 - 2 2 - 2 - 2 2 = span 1 - 1 0 , 1 0 1 eigenbasis : 1 1 1 , 1 - 1 0 , 1 0 1 (b) (10 points) Consider the discrete dynamical system ~x ( t + 1) = A~x ( t ) , ~ x 0 = ~x (0) = 2 1 2 . Using part (a), find a formula for ~x ( t ), for t 1. Solution : x 0 = 2 1 2 = C 1 1 1 1 + C 2 1 - 1 0 + C 3 1 0 1 so C 1 = 1 ,C 2 = 0 ,C 3 = 1 ~x ( t ) = C 1 λ t 1 ~ v 1 + C 2 λ t 2 ~ v 2 + C 3 λ t 3 ~ v 3 = 0 t ~ v 1 + 2 t ~ v 3 = 0 t 1 1 1 + 2 t 1 0 1 = 2 t 1 0 1
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(c) (5 points) Determine an invertible matrix S and a diagonal matrix D such that D = S - 1 AS . (You do not need to compute S - 1 !). Solution : D = λ 1 0 0 0 λ 2 0 0 0 λ 3 = 0 0 0 0 2 0 0 0 2 S = ± ~ v 1 ~ v 2 ~ v 3 ² = 1 - 1 1 1 1 0 1 0 1 2. Let V = span { e t ,te t ,t 2 e t } . You are told that these three functions form a basis B of V . Let T : V -→ V be given by T ( f ) = f 0 ( t ) - f ( t ) . (a)
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This note was uploaded on 12/25/2009 for the course MATH 1920 taught by Professor Pantano during the Spring '06 term at Cornell University (Engineering School).

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prelim2_09Sol - Prelim 2 Solutions Math 2940 November 19,...

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