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Algebra2008aug - Algebra Preliminary Examination Print name...

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Unformatted text preview: Algebra Preliminary Examination, August 19, 2008 Print name: Show your work, give reasons for your answers, provide all necessary proofs and counterexamples. There are 10 problems on 15 pages. Each problem is worth 10 points for a total of 100 points. Please check that you have a complete exam and print your name on each page. Problem 1 Problem 6 Problem 2 Problem 7 Problem 3 Problem 8 Problem 4 Problem 9 Problem 5 Problem 10 Total Except for your name, do NOT write on this page 1 Print name: 1. Let R 2 be the Euclidean plane with the standard basis e 1 = 1 , e 2 = 1 . (a) (3 points) The counterclockwise rotation about the origin through an angle α, where- ∞ < α < ∞ , is a linear operator on R 2 . Find the representation matrix of this linear operator with respect to the standard basis { e 1 ,e 2 } . (b) (5 points) The orthogonal reflection about a line through the origin is a linear operator on R 2 . Denote by L 1 the x-axis, and by L 2 the line obtained by rotating L 1 about the origin through a counterclockwise angle θ, where 0 < θ < π/ 2. Denote by r i the orthogonal reflection about...
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