prelim2_fa06practice

# prelim2_fa06practice - Sample questions for Prelim 2 Math...

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Sample questions for Prelim 2 Math 294 Fall 2006 There is no exact match for the current prelim since this year the second prelim comes about two weeks later. 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.

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