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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 ]? The coordinates of these vectors in the standard basis are (1 , 3 , 2) T , (2 , 1 , 1) T and (7 , 4 , 5) T . Since the matrix 1 2 7 3 1 4 2 1 5 is invertible (row reduce it, or take its determinant, which is 6), those three coordinates form a basis for R 3 , so the original vectors form a basis for R 2 [ t ]. b) Find the changeofbasis matrices P EB and P BE . The matrix P EB is made from the coordinates of the B vectors in the E basis, and is 1 1 2 1 3 1 1 . P BE is the inverse of this matrix, namely 1 2 1 1 5 1 2 . c) Find [ x ] B , where x = 1 + 10 t + 100 t 2 . [ x ] B = P BE [ x ] E = 1 2 1 1 5 1 2 1 10 100 = 1 108 205 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 . Since L ( e 1 ) = e 4 , L ( e 2 ) = e 2 , L ( e 3 ) = e 3 and L ( e 4 ) = e 1 we have [ L ] E = 1 1 1 1 . b) Find a basis for the eigenspace E 1 . 1 We are looking for the null space of L ( 1) I = 1 1 1 1 . This is one equation in 4 unknowns, namely x 1 = x 4 . The three free variables are x 2 , 3 , 4 , and our basis vectors are (0 , 1 , , 0) T , (0 , , 1 , 0) T , and ( 1 , , , 1) T . These are eigenvectors of the matrix [ L ] E and correspond to the eigenvectors parenleftbigg 1 parenrightbigg , parenleftbigg 1 parenrightbigg , parenleftbigg 1 1 parenrightbigg of the operator L . 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 . You do not need to compute the eigenvectors....
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This note was uploaded on 04/10/2011 for the course M 346 taught by Professor Radin during the Spring '08 term at University of Texas.
 Spring '08
 RAdin
 Linear Algebra, Algebra, Vectors

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