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Practice Prelim 2

# Practice Prelim 2 - Sample questions for Prelim 2 Math 2940...

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Sample questions for Prelim 2 Math 2940 Fall 2010 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 R 4 be the linear transformation defined by T a b c = a - 2 b + 3 c 3 a + 2 b + c a + 2 b - c a + c . (a) Find a matrix A so that T a b c = A a b c . (b) Find the dimension and a basis for C ( A ). (c) Find the dimension and a basis for N ( A ). 2. In each of the following, you are given a vector 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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Practice Prelim 2 - Sample questions for Prelim 2 Math 2940...

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