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Midterm 2-Fall 2005 - AMS 510.01 Fall 2005 Midterm#2 Name...

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AMS 510.01 Fall 2005, Midterm #2 Name: Student ID: Score: /100 ( + /15 bonus) 1. Indicate whether each of the following statements is true or false. (2 points each). (a) Every linear transformation may be represented as a matrix. (b) An orthonormal basis can always be defined for any vector space with a defined inner product. (c) If all entries of a real, symmetric, n -square matrix, A , are positive, then x T A x > 0 for any n × 1 vector x = 0. (d) The following matrix is invertible: 0 3 0 2 7 1 0 9 6 9 2 4 0 7 0 0 (e) The eigenvalue decomposition of a matrix, A (given by A = PDP - 1 where D is diagonal) is unique. (f) An n -square matrix always has n real eigenvalues if and only if the matrix is invertible. (g) All functions that are continuous everywhere in [a,b] are differentiable everywhere in (a,b). (h) A infinite series, i =1 u i , converges if and only if the corresponding sequence { u i } converges to zero ( i.e. lim i →∞ u i = 0). (i) lim x →∞ x 2 sin x = .
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