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Unformatted text preview: M346 Final Exam, December 15, 2009 1) The matrix A = 1 3 2 5 2 1 1 1 1 2 3 3 3 5 7 rowreduces to B = 1 4 / 11 1 13 / 11 1 10 / 11 . a)Find all solutions to A x = 0. These are the same as the solutions to B x = 0, namely all multiples of (4 / 11 , 13 / 11 , 10 / 11 , 1) T , or equivalently all multiples of (4 , 13 , 10 , 11) T . b)Find a basis for the column space of A . Since there are pivots in the first three columns of B , the first three columns of A for a basis. That is, the answer is 1 2 3 , 3 1 1 3 , 2 1 2 5 . c)In R 3 [ t ], let V be the span of the vectors { 1 + 2 t + 3 t 3 , 3 t + t 2 + 3 t 3 , 2+ t +2 t 2 +5 t 3 , 5 t +3 t 2 +7 t 3 } . What is the dimension of V ? Find a basis for V . If you express things in coordinates with respect to the standard basis { 1 , t, t 2 , t 3 } , this becomes the same problem as (b). V is 3dimensional, as a basis consists of those polynomials whose coordinates are the answer to (b), namely { 1+2 t +3 t 3 , 3 t + t 2 +3 t 3 , 2+ t +2 t +2+5 t 3 } . Note that the answer is NOT a matrix or a list of column vectors. Those are just the coordinates of the answer, not the answer itself. 2. a) Find the eigenvalues of 3 5 16 4 3 11 15 1 4 1 2 . You do not need to find the eigenvectors. The matrix is block triangular, with an upper left 1 1 block and a lower right 3 3 block. The 3 3 block is itself block triangular, with an upper left 2 2 piece parenleftbigg 3 11 15 1 parenrightbigg and a lower right 1 1 piece. The rows of the 2 2 piece sum to 14, and the trace is 2, so that piece has eigenvalues 14 and 12, and the whole matrix has eigenvalues 3 , 14 , 12 , 2. b) Find the eigenvalues and eigenvectors of parenleftbigg 3 8 2 3 parenrightbigg ....
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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 at Austin.
 Spring '08
 RAdin
 Linear Algebra, Algebra

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