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midterm2solutions

midterm2solutions - Midterm 2 Math 33A Fall 2007 Name UCLA...

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Midterm 2, Math 33A, Fall 2007 November 19, 2007 Name: UCLA ID: Section (circle the one you regularly attend, as this is where the exam will be returned): 2A Jack Buttcane Tuesday 2B Jack Buttcane Thursday 2C Judah Jacobson Tuesday 2D Judah Jacobson Thursday Directions: Fill in your name and circle your section above. Do not turn the page until instructed to do so. You have 50 minutes to complete the exam. No outside materials are allowed; use only your brain and a writing instrument. There are 3 problems; each is worth 10 points total. Extra scratch paper is included. If your work on a problem appears on a different page, indicate clearly where it may be found. Show all the necessary steps involved in finding your solutions, unless otherwise instructed. In the interest of us not losing pages of your exam, please refrain from detaching pages from the exam. Good luck. Problem Score 1 2 3 Total 1
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1. Consider the following orthonormal basis of R 3 : B = { ~v 1 ,~v 2 ,~v 3 } where ~v 1 = 1 3 1 2 2 , ~v 2 = 1 3 - 2 2 - 1 , ~v 3 = 1 3 - 2 - 1 2 . Let T be the linear transformation which rotates vectors 90 degrees counterclockwise around the ~v 1 axis, as viewed from the tip of the vector ~v 1 looking in toward the origin. a. (3 pts) Compute the matrix of T with respect to the basis B . b. (7 pts) Compute the matrix of T with respect to the standard basis. Solution. a. Drawing a picture of the three vectors ~v 1 ,~v 2 ,~v 3 , we see that T ( ~v 1 ) = ~v 1 , T ( ~v 2 ) = ~v 3 , T ( ~v 3 ) = - vv 1 . Hence [ T ( ~v 1 )] B = 1 0 0 , [ T ( ~v 2 )] B = 0 0 1 , [ T ( ~v 3 )] B = 0 - 1 0 .
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