finweb2005 - M346 Final Exam, December 15, 2005 1. In R 2 [...

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Unformatted text preview: M346 Final Exam, December 15, 2005 1. In R 2 [ t ], consider the standard basis E = 1 ,t,t 2 and the alternate basis B = 1- t + 3 t 2 , 2 t- t 2 ,- t + t 2 . a) Are the vectors b 1 = 1 + 3 t + 2 t 2 , b 2 = 2 + t + t 2 , b 3 = 7 + 4 t + 5 t 2 linearly independent? Do they span R 2 [ t ]? b) Find the change-of-basis matrices P EB and P BE . c) Find [ x ] B , where x = 1 + 10 t + 100 t 2 . 2. On M 2 , 2 , consider the linear transformation L parenleftbigg a b c d parenrightbigg = parenleftbigg d- b- c a parenrightbigg . a) Find the matrix of L with respect to the standard basis E = braceleftbiggparenleftbigg 1 parenrightbigg , parenleftbigg 1 parenrightbigg , parenleftbigg 1 parenrightbigg , parenleftbigg 1 parenrightbiggbracerightbigg . b) Find a basis for the eigenspace E- 1 . 3. a) Find the eigenvalues of the matrix 3 2 3 1 4 1- 2 3 2 1 7 1 4 1 4 1 2 1 1 2 1 1 2 ....
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finweb2005 - M346 Final Exam, December 15, 2005 1. In R 2 [...

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