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Unformatted text preview: M346 Final Exam 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 . b) Find the matrix of L 2 relative to the same basis. 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? 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? Problem 3. Let A = 1 5 3 4 4 3 . Which of the following are Hermitian? Which are unitary? Which are both? Which are neither? a) A b) A + I c) e A d) e iA Problem 4. In R 4 with the standard inner product, consider the vectors b 1 = (1 , , , 1) T , b 2 = (1 , 2 , 2 , 1) T , b 3 = (2 , 1 , 1 , 0) T , b 4 = (1 , 3 , 5 , 7) T ....
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 Spring '08
 RAdin
 Linear Algebra, Algebra, Equations, Transformations, Matrices, Vector Space, Fundamental physics concepts, Hilbert space, Dirichlet boundary conditions

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