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Unformatted text preview: Mathematics 54W Professor A. Ogus Spring, 2005 Midterm Solutions—April 07, 2005 Work each problem on a separate sheet of paper. Be sure to put your name, your section number, and your GSI’s name on each sheet of paper. Also, at the top of the page, in the center, write the problem number, and be sure to put the pages in order. Write clearly: explanations (with complete sentences when appropriate) will help us understand what you are doing. Note that there are problems on the back of this sheet, for a total of five problems. 1. (5 pts) Let A := 1 1 1 , let W be the column space of A , and let Y := 1 1 3 . (a) (5 pts) Use the Gram-Schmidt process to find an orthogonal basis for W . w 1 = 1 1 ,w 2 = 1- 1 . (b) (5 pts) Find the orthogonal projection Y of Y on W . Y := ( Y | w 1 ) ( w 1 | w 1 ) + ( Y | w 2 ) ( w 2 | w 2 ) = 4 2 w 1 +- 2 2 w 2 = 2 w 1- w 2 = 1 3 ....
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This note was uploaded on 03/08/2010 for the course MATH 1A taught by Professor Wilkening during the Spring '08 term at Berkeley.
- Spring '08