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Unformatted text preview: Prelim 2 Math 294 Fall 2005 no calculators or notes answer + reason = credit there are 2 questions on each side of this page; 25 points each there are some pieces of formulas and definitions on the bottom of the second side 1. Let T : R 3 P 4 be defined by T a b c = p ( x ) = ( a- 2 b +3 c )+(3 a +2 b + c ) x +( a +2 b- c ) x 2 +( a + c ) x 4 (a) (9 points) Find the dimension and a basis for im( T ). (b) ( 8 points) Find the coordinates of the polynomial p ( x ) above in terms of your basis for im( T ). (c) (8 points) Find the dimension and a basis for ker( T ). 2. (a) In each of the following, you are given a linear space V and a subset W V . Decide whether W is a subspace of V , and prove your answer is correct. i. (7 points) V is the space R 2 2 of all 2 2 matrices, and W is the set of 2 2 matrices A such that A 2 = A . ii. (7 points) V is the space of differentiable functions, and W is the set of those differentiable functions that satisfy...
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- Spring '05