final-practice - Math 2940 Practice problems for nal Dec...

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Math 2940 – Practice problems for final – Dec 2009 On the final exam : The formula sheet on the last page will not be present. Instead, you may bring into the exam one 2-sided, 8.5 by 11 inch sheet of paper, that you have written yourself. 1. Let A = 1 2 - 1 3 - 3 2 4 0 0 1 4 8 - 2 6 - 5 (a) Find a basis for the image of A . (b) Find a basis for the kernel of A . (c) What is the rank of A ? 2. Consider the linear space C [ - 1 , 1] with inner product defined by < f, g > = Z 1 - 1 f ( t ) g ( t ) dt. (a) Compute the lengths of the functions f 1 ( t ) = e t , and f 2 ( t ) = e - t . (b) Are f 1 ( t ) and f 2 ( t ) orthogonal? Justify your answer. (c) Find the best least squares approximation to the constant func- tion g ( t ) = 1 on [ - 1 , 1] by a function of the form c 1 f 1 ( t )+ c 2 f 2 ( t ), where c 1 and c 2 are real numbers. 3. Let V be the subspace of P 3 consisting of all f ( t ) such that f (1) = 0 and f 0 (1) = 0 . (a) Find a basis for V . (b) Write 4 t 3 - 7 t 2 +2 t +1 as a linear combination of your basis from part (a).
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