finsol2005 - M346 Final Exam 1 In R2[t consider the...

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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 change-of-basis 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 0 0 - 1 2 - 1 3 - 1 1 . P BE is the inverse of this matrix, namely 1 0 0 - 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 0 0 - 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 0 0 0 parenrightbigg , parenleftbigg 0 1 0 0 parenrightbigg , parenleftbigg 0 0 1 0 parenrightbigg , parenleftbigg 0 0 0 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 = 0 0 0 1 0 - 1 0 0 0 0 - 1 0 1 0 0 0 . b) Find a basis for the eigenspace E - 1 . 1
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We are looking for the null space of L - ( - 1) I = 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 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 , 0) T , (0 , 0 , 1 , 0) T , and ( - 1 , 0 , 0 , 1) T . These are eigenvectors of the matrix [ L ] E and correspond to the eigenvectors parenleftbigg 0 1 0 0 parenrightbigg , parenleftbigg 0 0 1 0 parenrightbigg , parenleftbigg - 1 0 0 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 0 0 4 1 4 1 0 0 0 2 1 0 0 0 0 1 2 1 0 0 0 0 1 2 . You do not need to compute the eigenvectors. This is block-triangular. The upper left 2 × 2 block has eigenvalues 3 ± 2 i , the middle block has eigenvalue 4, and the lower right 3 × 3 block has eigenvalues 2 , 2 ± 2. (The upper block is of the form parenleftbigg a - b b a parenrightbigg , which always has eigenvalues a ± bi , and the lower block should be familiar from homework. It also isn’t too hard to compute directly.)
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