math54-Spring07-final-Ribet-soln

math54-Spring07-final-Ribet-soln - Math 54 Final Exam...

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Math 54 Professor K. A. Ribet Final Exam May 18, 2007 This was a 3-hour exam, 12:30–3:30PM. There were 60 points for eight questions. Prob- lem 1 consisted of 11 true–false questions, each worth 1 point. Problems 2–8 had point values 6, 8, 9, 6, 6, 8 and6. I think that it was a successful exam: people seemed happy, and there weren’t problems that were obviously ambiguous. The only misprint that I know about was in the last problem, where “diagonal” should replace “diagonalizable.” We announced this in the exam room. 1. For each statement below, write TRUE or FALSE to the left of the statement. You are not required to justify your reasoning: If A is a square invertible matrix, then A and A - 1 have the same rank. True: the rank is the size of the matrix A in both cases. If A is an m × n matrix and if b is in R m , there is a unique x R n for which Ax - b is smallest. False: for example, A could be the 0-matrix and b could be 0. Then the length is smallest (namely 0) for all x . If A is an n × n matrix, and if v and w in R n satisfy Av = 2 v , Aw = 3 w , then v · w = 0 . False: it’s not true in general that eigenvectors for different eigenvalues are perpendicular. We proved this for symmetric matrices, however. If the dimensions of the null spaces of a matrix and its transpose are equal, then the matrix is square. True by the rank-nullity theorem, since a matrix and its transpose have the same rank. If A is a 2 × 2 matrix, then - 1 cannot be an eigenvalue of A 2 . False. For example, if A = 0 1 - 1 0 , then A 2 = - I 2 . I liked the linear algebra portion of this course more than the differential equations portion. This was supposed to be a “free point,” but students who gave no answer probably won’t get their empty answer marked correct.
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If four linearly independent vectors lie in Span( { w 1 , . . . , w t } ) , then t must be at least 4.
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