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prelim2practice - Sample questions for Prelim 2 Math 2940...

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Sample questions for Prelim 2 Math 2940 Fall 2009 This represents relevant questions that have appeared on previous prelims and finals. The overall length is not representative of a single prelim. 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) Find the dimension and a basis for im( T ). (b) Find the coordinates of the polynomial p ( x ) above in terms of your basis for im( T ). (c) Find the dimension and a basis for ker( T ). 2. 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. (a) 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 . (b) V is the space of differentiable functions, and W is the set of those differentiable functions that satisfy f (3) = 0. 3. (a) Find a quadratic function of the form f ( x ) = c + dx 2 that best fits the data ( x, y ) = ( - 1 , 1) , (0 , 1) , (1 , 2) in the least squares sense.
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