Midterm 2-Fall 2005-Solutions

Midterm 2-Fall 2005-Solutions - AMS 510.01 Fall 2005,...

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Unformatted text preview: AMS 510.01 Fall 2005, Midterm #2 Solutions 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. True. The matrix consisting of the coordinate (column) vectors of the transformed elements of any basis gives a matrix representation. The representation always exists, but will vary depending on the bases used for the source and target spaces. (b) An orthonormal basis can always be defined for any vector space with a defined inner product. True. The GramSchmidt orthonormalization algorithm can be used to convert any basis into an orthonormal basis, so long as an inner product is defined. (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 6 = 0. False. ~x T A ~x > 0 defines a positive definite matrix; having all positive entries is not a sufficient condition for a matrix to be positive definite. (d) The following matrix is invertible: 3 0 2 7 1 0 9 6 9 2 4 0in- - - 0 7 0 0 True. The determinant is non-zero: 0 3 0 2 7 1 0 9 6 9 2 4 0 7 0 0 = (7) 0 0 2 7 0 9 6 2 4 = (7)(2) 7 0 6 2 = 14(7 2- 6) 6 = 0 (e) The eigenvalue decomposition of a matrix, A (given by A = PDP- 1 where D is diagonal) is unique. False. While the eigenvalues of a matrix are unique, any basis for each associated eigenspace may be used to create the matrix P . Even if all eigenspaces have dimen- sionality of one, any scalar multiple of each eigenvector may be used. (f) An n-square matrix always has n real eigenvalues if and only if the matrix is invertible. False. There is no relation between whether a matrix is invertible and whether is it diagonalizable. (g) All functions that are continuous everywhere in [a,b] are differentiable everywhere in (a,b). False. A function may be continuous on an interval while not being differentiable. For example, y = | x | is continuous on any interval in (- , ), but is not differentiable at x = 0. (h) A infinite series, i =1 u i , converges if and only if the corresponding sequence { u i } con- verges to zero ( i.e. lim i u i = 0). False. lim i u i = 0 is a necessary condition for i =1 u i to converge, but it is not sufficient. For example, i =1 1 i diverges (to ) while the sequencue { 1 i } converges to zero. (i) lim x x 2 sin x = . False. While x 2 increases monotonically to with increasing x , the term sin x oscillates continuously between 1 regardless of how large x becomes. As a result, the product oscillates between ever increasing positive and negative values....
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This note was uploaded on 02/09/2010 for the course AMS 510 taught by Professor Feinberg,e during the Fall '08 term at SUNY Stony Brook.

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Midterm 2-Fall 2005-Solutions - AMS 510.01 Fall 2005,...

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