ps6 - Problem Set 6 Math 115A/5 — Fall 2011 Due Friday...

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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 finite-dimensional 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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ps6 - Problem Set 6 Math 115A/5 — Fall 2011 Due Friday...

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