ef11k - MAS 3105 Final Exam and Key April 28, 2011 Prof. S....

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MAS 3105 April 28, 2011 Final Exam and Key Prof. S. Hudson 1) Suppose L rotates each vector in R 2 by 45 degrees clockwise. Find the matrix repre- sentation of L (standard basis). Hint: cos(45) = p 1 / 2. 2) [20 pts] True-False. You can assume the matrices are all square. All the eigenvectors of a nilpotent matrix are 0. If eigenvectors x 1 and x 2 correspond to λ 1 6 = λ 2 then x 1 x 2 . If A is similar to B then they have the same rank. The Google PageRank algorithm is a Markov process. A R 3 × 3 such that (3 , 1 , 1) Row( A ) and (1 , 1 , 3) T N ( A ). If L : V W is linear, then ker( L ) L ( V ). We used a basis of 3 eigenvectors to solve the Rabbit problem. For all 3 × 3 matrices, rank ( AB ) rank ( B ). The normal equations are AA T x = A T b . Overdetermined systems are usually inconsistent. 3) Choose ONE [from Ch 6.4 HW] a) Show that if U is unitary and x C n , then || U x || = || x || . b) Show that if
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ef11k - MAS 3105 Final Exam and Key April 28, 2011 Prof. S....

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