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Unformatted text preview: Problem Set 6 Math 115A/5 — Fall 2011 Due: Friday, November 18 Problem 1 (4.4.6) Prove that if M ∈ M n × n ( F ) can be written in the form M = A B C , where A and C are square matrices, then det( M ) = det( A ) · det( C ). Problem 2 (5.1.3) For each of the following matrices A ∈ M n × n ( F ), determine all eigenvalues of A ; for each eigenvalue λ of A , find the set of eigenvectors corresponding to λ ; if possible, find a basis for F n consisting of eigenvectors of A ; if succesful in finding such a basis, determine an invertible matrix Q and a diagonal matrix D such that Q 1 AQ = D . (a) A = 1 2 3 2 for F = R . (b) A =  2 3 1 1 1 2 2 5 for F = R . (c) A = i 1 2 i for F = C . (d) A = 2 0 1 4 1 4 2 0 1 for F = R . Problem 3 (5.1.7a) Let T be a linear operator on a finitedimensional vector space V . We define the determinant of T , denoted det( T ), as follows: Choose any ordered basis β for V , and define det( T ) = det([ T ] β ). Prove that the preceeding definition is independent of the choice of ordered basis for V ....
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 Fall '08
 Kong,L
 Math, Linear Algebra, Determinant, Matrices, Det, linear operator, finitedimensional vector space

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