Unformatted text preview: M ATH 223 P ROBLEM S ET 7 D UE : 23 October 2007 When you hand in this problem set, please indicate on the top of the front page how much time it took you to complete. Reading. 2.72.10. Problems from the book: the starred problems will be graded. 2.7.2, 2.7.3, 2.7.6 2.8.2, 2.8.3, 2.8.9, 2.8.13 2.9.6 Additional problems: 1. Let f : R2 R3 and g : R3 R2 be defined by x2 - 3y a x 2 f = 2xy + y and g b = y x + y - 17 c a - b + c2 abc . Both by substitution and by the chain rule, compute the Jacobian of g f and of f g. 2. Suppose that A is an invertible n n matrix. Show that is an eigenvalue of A if 1 and only if is an eigenvalue of A-1. 3. Suppose A and B are n n matrices. Show that AB and BA have the same eigenvalues. 4. Suppose that f : R R is differentiable and satisfies |f (x)| M for some real number M and for all x R. Show that f as Lipschitz ratio M. ...
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This homework help was uploaded on 02/24/2008 for the course MATH 2230 taught by Professor Holm during the Fall '07 term at Cornell.
- Fall '07