Scott Armstrong math 54 midterm 2 spring 2008

# Scott Armstrong math 54 midterm 2 spring 2008 - Math 54...

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Math 54 - Midterm #2 July 30, 2008, 08:00-10:00 Name: This is a closed book, closed notes exam. Calculators are not allowed. You have two hours to complete the exam. To receive full credit, write legibly, show your work and write proofs in complete sentences. If you need more space, use the back of the page of the problem on which you are working. Problem Points Your Score 1 20 2 20 3 20 4 20 5 20 6 20 Total 120

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1. Let A = - 1 3 0 0 - 2 0 2 6 - 2 (a) Find the eigenvalues of A . (6 points) (b) Find bases for the eigenspaces of A . (8 points) (c) Is A diagonalizable? If so, diagonalize A . (6 points)
2. Let W = span 5 1 - 2 , 0 4 2 and y = - 6 12 6 . (a) Write y as the sum of a vector ˆ y W and a vector z W . (10 points) (b) What is the distance from y to W ? Explain. (5 points) (c) Find a basis for W . (5 points)

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3. Let B = { 1 ,t,t 2 } and C = { 1 - t, 1 + t,t 2 + 1 } , which are bases for the vector space
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Scott Armstrong math 54 midterm 2 spring 2008 - Math 54...

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