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HW7 - Math 3013 Homework Set 7 Problems from 3.3(pgs...

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Math 3013 Homework Set 7 Problems from § 3.3 (pgs. 211-212 of text): 1,3,5,7,9,11 Problems from § 3.4 (pgs. 226-229 of text): 1,5,7,16,20,26 1. (Problems 1,3,5,7,9,11 in § 3.3) Find the coordinate vector of the given vector relative to the indicated ordered basis. (a) (Problem 3.3.1) [ - 1 , 1] R 2 , relative to ([0 , 1] , [1 , 0]) (b) (Problem 3.3.3) [4 , 6 , 2] R 3 , relative to ([2 , 0 , 0] , [0 , 1 , 1] , [0 , 0 , 1]) (c) (Problem 3.3.5) [3 , 13 , - 1] R 3 , relative to ([1 , 3 , - 2] , [4 , 1 , 3] , [ - 1 , 2 , 0]) (d) (Problem 3.3.7) x 3 + x 2 - 2 x + 4 in P 3 , relative to ( 1 , x 2 , x, x 3 ) (e) (Problem 3.3.9) x + x 4 in P 4 , relative to ( 1 , 2 x - 1 , x 3 + x 4 , 2 x 3 , x 2 + 2 ) (f) (Problem 3.3.11) x 3 - 4 x 2 + 3 x + 7 relative to B = ( x - 2) 3 , ( x - 2) 2 , x - 2 , 1 2. Let F be the vector space of functions f : R R . Determine whetther the following functions T are linear transformations. (a) (Problem 3.4.1) T : F R : T ( f ) = f ( - 4) . (b) (Problem 3.4.5) T : F F : T ( f ) = - f 3. (Problem 3.4.7) Let C be the space of all continuous functions mapping R to R , and let T : C C be defined by T ( f ) = x 1 f ( t ) dt . If possible give three different functions in ker ( T ). 4. (Problem 3.4.16) Let V and V be vector spaces having ordered bases B = ( b 1 , b 2 , b 3 ) and B = ( b 1 , b 2 , b 3 , b 4 ), respectively. Let T : V V be a linear transformation such that T ( b 1 ) = 3 b 1 + b 2 + 4 b 3 + b 4 T ( b 2
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