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finweb2004 - M346 Final Exam 1 On R3[t let L be the linear...

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M346 Final Exam, December 11, 2004 1. On R 3 [ t ], let L be the linear operator that shifts a function to the left by one. That is ( L p )( t ) = p ( t + 1). Find the matrix of L relative to the standard basis { 1 , t, t 2 , t 3 } 2. a) Find the eigenvalues of the following matrix. You do NOT have to find the eigenvectors. 3 2 1 1 2 3 1 2 3 4 5 6 1 1 1 7 8 9 0 0 0 4 0 0 0 0 0 3 3 4 0 0 0 5 4 3 b) Find the eigenvalues AND eigenvectors of the matrix parenleftbigg 1 4 1 0 parenrightbigg . Also find the eigenvalues AND eigenvectors of the matrix parenleftbigg 1 - 4 1 0 parenrightbigg . 3. A 3 × 3 matrix A has eigenvalues 2, 1 and - 1 and corresponding eigen- vectors b 1 = (1 , 2 , 3) T , b 2 = (1 , 1 , - 1) T and b 3 = ( - 5 , 4 , - 1) T . a) Decompose (36 , 1 , 34) T as a linear combination of b 1 , b 2 and b 3 . b) If d x /dt = A x and x (0) = (36 , 1 , 34) T , what is x ( t )? [You do NOT need to compute A to do this.] c) Is A Hermitian? Why or why not? Is A unitary?
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