mat67-Homework9

mat67-Homework9 - MAT067 University of California, Davis...

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MAT067 University of California, Davis Winter 2007 Homework Set 9: Exercises on Orthogonality and Diagonalization Directions : Please work on all of the problems and submit your solutions to the Calcula- tional Problems 1 and 2, and Proof-Writing Problems 3 and 6 at the beginning of lecture on March 9, 2007. As usual, we are using F to denote either R or C .W ea l sou s e , ·i to denote an arbitrary inner product and k·k to denote its associated norm. The term “o.n.” means orthonormal . A denotes the conjugate transpose of a matrix A C n × n . 1. Consider R 3 with two orthonormal bases: the canonical basis e =( e 1 ,e 2 3 )andthe basis f f 1 ,f 2 3 ), where f 1 = 1 3 (1 , 1 , 1) 2 = 1 6 (1 , 2 , 1) 3 = 1 2 (1 , 0 , 1) . (a) Find the matrix, S , of the change of basis transformation such that [ v ] f = S [ v ] e , for all v R 3 , where [ v ] b denotes the column vector with the coordinates of the vector v in the basis b . (b) Find the canonical matrix, A , of the linear map T ∈L ( R 3 ) with eigenvectors f 1 2 3 and eigenvalues 1 , 1 / 2 ,
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This homework help was uploaded on 04/08/2008 for the course MATH 67 taught by Professor Schilling during the Fall '07 term at UC Davis.

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mat67-Homework9 - MAT067 University of California, Davis...

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