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**Unformatted text preview: **M346 Final Exam Soutions December 13, 2000 Problem 1. Consider the vector space M 2 , 2 of 2 × 2 matrices, let B = 2 3 5 . Con- sider the linear transformations L 1 ( A ) = AB and L 2 ( A ) = BA . a) Find the matrix of L 1 relative to the basis 1 , 1 , 1 , 1 . We compute: L 1 b 1 = 1 2 3 5 = 2 = 2 b 2 L 1 b 2 = 1 2 3 5 = 3 5 = 3 b 1 + 5 b 2 L 1 b 3 = 1 2 3 5 = 2 = 2 b 4 L 1 b 4 = 1 2 3 5 = 3 5 = 3 b 3 + 5 b 4 so we have [ L 1 ] B = 3 2 5 3 2 5 b) Find the matrix of L 2 relative to the same basis. We compute: L 2 b 1 = 2 3 5 1 = 3 = 3 b 3 L 1 b 2 = 2 3 5 1 = 3 = 3 b 4 L 1 b 3 = 2 3 5 1 = 2 5 = 2 b 1 + 5 b 3 L 1 b 4 = 2 3 5 1 = 2 5 = 2 b 2 + 5 b 4 so we have [ L 2 ] B = 2 2 3 5 3 5 Problem 2. Let A = 1 4- 4 3 7 . a) Consider the equations x ( n ) = A x ( n- 1), with A as above. What are the stable and unstable modes? What is the dominant eigenvector? The eigenvalues are λ 1 = 3 / 4 and λ 2 =- 7 / 4, with eigenvectors b 1 = (3 , 7) T and b 2 = (1 ,- 1) T . The first mode is stable since | λ 1 | < 1, while the second is unstable since | λ 2 | > 1. The dominant eigenvector is λ 2 =- 7 / 4. b) Consider the equations ˙ x ( t ) = A x ( t ), with A as above. What are the stable and unstable modes? What is the dominant eigenvector? Now the question is whether the real part of λ is positive or negative....

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