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Unformatted text preview: CAAM 336 Â· DIFFERENTIAL EQUATIONS Problem Set 2 Posted Wednesday 1 September 2010. (Corrected 6 Sept. 2010.) Due Wednesday 8 September 2010, 5pm. 1. [27 points] Consider the following sets of functions. Demonstrate whether or not each is a vector space (with addition and scalar multiplication defined in the obvious way). (a) { x âˆˆ R 2 : x 2 = x 3 1 } (b) { x âˆˆ R 3 : x 1 + 2 x 2 + 3 x 3 = 0 } (c) { f âˆˆ C [0 , 1] : f ( x ) â‰¥ 0 for all x âˆˆ [0 , 1] } (d) { f âˆˆ C [0 , 1] : max x âˆˆ [0 , 1] f ( x ) â‰¤ 1 } (e) { f âˆˆ C 1 [0 , 1] : f (0) = 0 } (f) { f âˆˆ C 2 [0 , 1] : f 00 ( x ) = 0 for all x âˆˆ [0 , 1] } 2. [18 points] (a) Suppose that f : R 2 â†’ R 2 is linear. Prove there exists a matrix A âˆˆ R 2 Ã— 2 such that f is given by f ( u ) = Au . Hint: Each u = u 1 u 2 âˆˆ R 2 can be written as u = u 1 e 1 + u 2 e 2 , where e 1 = 1 , e 2 = 1 . Since f is linear, we have f ( u ) = u 1 f ( e 1 ) + u 2 f ( e 2 ). Your formula for the matrix A may include the vectors f ( e 1 ) and f...
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This note was uploaded on 01/21/2012 for the course CAAM 330 taught by Professor Tompson during the Fall '09 term at UVA.
 Fall '09
 Tompson

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