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ef09k - MAS 3105 Final Exam and Key Prof S Hudson 1a[each...

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MAS 3105 June 18, 2009 Final Exam and Key Prof. S. Hudson 1a) [each part of problem 1 is 5 points]. In the Rabbit example, we studied L : R 2 R 2 where L ([ a, b ] T ) = [ a + b, 2 a ] T . Find the matrix representation A of L (w.r.t. the std basis of R 2 ). 1b) We found two eigenvectors of A , x 1 = [1 , 1] T and x 2 = [1 , - 2] T , which form a basis X of R 2 . Find the corresponding eigenvalues. 1c) Find the matrix representation B of L w.r.t. the basis X . 1d) Find an eigenvector for B . Is it also an eigenvector of A ? Explain briefly. 1e) Let x = [2 , 3] T X (the coordinates are wrt the eigenvector basis, X ). Find [ L ( x ] S , in standard coordinates. 2) [20 pts] True-False. You can assume the matrices are all square. A T A and AA T always have the same rank. If A and B are unitary then AB is unitary. If A is singular then AB is singular. If A H = - A then A is normal. If A is unitary then A is not defective. If A is Hermitian and unitary then A 2 = I . If U , V are subspaces of R n , and U V then V U . If A is unitary and x C n then || A x || = || x || . If λ is an eigenvalue of a unitary matrix then | λ | = 1. If λ is an eigenvalue of a unitary matrix then λ is real. 3) [10pts] Suppose A is a 4x4 matrix and det A = 3. Find det (adj( A )).
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