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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 . We also use · , · to denote an arbitrary inner product and · 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 , e 3 ) and the basis f = ( f 1 , f 2 , f 3 ), where f 1 = 1 3 (1 , 1 , 1) , f 2 = 1 6 (1 , 2 , 1) , f 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 , f 2 , f 3 and eigenvalues 1 ,

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

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