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Unformatted text preview: Second mid-term exam for Math 204 June 22, 2000 Rules and regulations • This exam is due at noon on Wednesday, April 12. • The exam is open-book; we ask you not to consult books other than your textbook, or other people. We also ask that you do not use computing devices for the problems on the exam. • If you feel there’s a typo on the exam, or that a question is unclear, please e-mail your section leader right away–your promptness will benefit your fellow students. • Please take the time to write clearly and in complete sentences, especially when you are writing a proof. And don’t forget to check your arithmetic–you have plenty of time to make sure everything is right. • Remember to write and sign the honor pledge on the front of your exam. • Good luck and have a good time! Problems 1. Rotations and axes We define a rotation to be an orthogonal matrix which has determinant 1. a. Give an example of a 3 × 3 permutation matrix, other than the identity, which is a rotation. What are the eigenvalues of this matrix? What are the eigenvectors? b. Give an example of a 3 × 3 rotation A such that A~e 1 = ~e 1 , where ~e 1 is the standard basis element 1 . What are the eigenvalues of A ? What are the eigenvectors? 1 c. Give an example of a rotation of the form A = 2 / 7 a b 3 / 7 c d 6 / 7 e f . d. Here is a false statement: if A is a rotation, the eigenvalues of A are all ± 1. Here is a fake proof of the false statement. Suppose ~v is an eigenvector for A , with eigenvector λ . Then A~v = λ~v. By Strang’s theorem 3R (p.168), and the fact that A is orthogonal, we have k ~v k = k A~v k = k λ~v k = λ 2 k ~v k . Since ~v is an eigenvector, it is nonzero, so k ~v k is nonzero; dividing out, we get λ 2 = 1 which yields λ = ± 1. Identify the incorrect step in the fake proof, and explain why it is incorrect. Physically speaking, an axis of a rotation is a line which is left unchanged by the rotation....
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This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.
- Fall '04