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Unformatted text preview: M ATH 223 P ROBLEM S ET 4 D UE : 2 O CTOBER 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. 2.6, 1.5&1.6. Problems from the book: the starred problems will be graded. 2.6.2, 2.6.3, 2.6.5, 2.6.8 , 2.6.9. 1.5.2 , 1.5.3, 1.5.7, 1.5.10, 1.5.18. 1.5.24 . Additional problems: 1. Are the following subsets of Pol5(R) subspaces? If so, find a basis and compute the dimension. (a) f Pol5(R) f(-x) = -f(x) ; (b) f Pol5(R) f(-x) = (f (x))2 ; and (c) f Pol5(R) f(2) = 0 . 2. Given a basis - , , . . . , - of Rn, the Gramm-Schmidt algorithm produces a new v1 vn orthonormal basis. Prove that this new basis is orthonormal! - - = v1 , u1 |- | v1 - - (- - )- u u v2 1 1 - = v2 u2 u u , |- - (- -1)-1| v2 v2 . . . k-1 - - - - - = vk - i=1 ( vk ui) ui . uk |- - k-1(- - )- | v v u u
k i=1 k i i 3. Suppose that f : R R is a continuous function. Consider the graph of f, graph(f) = (x, f(x)) R2 . Is it an open set in R2? Is it a closed set in R2? 4. Consider the set 1 x, sin R2 x > 0 . x Is it an open set in R2? Is it a closed set in R2? 5. Suppose that f : R R is a differentiable function. Given a real number c, must there exist a and b with a < c < b such that f(b) - f(a) f (c) = ? b-a Prove this, or give a counterexample. ...
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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 University (Engineering School).
- Fall '07