sol43 - Partial Solution Set, Leon 4.3 4.3.2 Let [u1 , u2 ]...

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Partial Solution Set, Leon § 4.3 4.3.2 Let [ u 1 , u 2 ] and [ v 1 , v 2 ] be ordered bases for R 2 , where u 1 = (1 , 1) T , u 2 = ( - 1 , 1) T , v 1 = (2 , 1) T , and v 2 = (1 , 0) T . Let L be the linear transformation defined by L ( x ) = ( - x 1 ,x 2 ) T , and let B be the matrix representing L with respect to [ u 1 , u 2 ]. { Note: B was actually part of problem 1 in this chapter. As usual, the first column of B is [ L ( u 1 )] U = (0 , 1) T , and the second column of B is [ L ( u ) 2 ] U = (1 , 0) T . } (a) Find the transition matrix S corresponding to the change of basis from [ u 1 , u 2 ] to [ v 1 , v 2 ]. Solution : The transition matrix in question is the one I’ve been calling T UV , i.e., S = V - 1 U = ± 0 1 1 - 2 ²± 1 - 1 1 1 ² = ± 1 1 - 1 - 3 ² . (b) Find the matrix A representing L with respect to [ v 1 , v 2 ] by computing A = SBS - 1 . Solution : First we find S - 1 = 1 2 ± 3 1 - 1 - 1 ² . Then it is a simple matter to determine that A = SBS - 1 = ± 1 0 - 4 - 1 ² . 4.3.3 Let L be the linear transformation on R 3 given by L ( x ) = (2 x 1 - x 2 - x 3 , 2 x 2 - x 1 - x 3 , 2 x 3 - x 1 - x 2 ) T , and let A be the matrix representing L with respect to the standard basis for R 3 . If u 1 = (1 , 1 , 0) T , u 2 = (1 , 0 , 1) T , and u 3 = (0 , 1 , 1) T , then [ u 1 , u 2 , u 3 ] is an ordered basis for R 3 .
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sol43 - Partial Solution Set, Leon 4.3 4.3.2 Let [u1 , u2 ]...

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