Spring 2003 - Buss' Class - Quiz 6

Spring 2003 - Buss' Class - Quiz 6 - 1 , u 2 , u 3 into an...

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Name: Thursday section time: Student ID: Math 20F - Linear Algebra - Spring 2003 Quiz #6 — May 29 (Do not discuss the quiz with students who haven’t taken it yet – until 8:00pm.) You must show your work in order to get credit for a problem. Label your answers clearly. 1. Let u 1 =(1 , 1 , 1 , 1) T , u 2 =(1 , - 1 , 1 , - 1) T and u 3 =(1 , 0 , - 1 , 0) T . Are these three vectors orthogonal? Are they orthonormal? ANSWER: They are orthogonal, but not orthonormal. 2. Let x =(1 , 2 , 3 , 4) T . Find the projection of x onto Span ( u 1 , u 2 , u 3 ). ANSWER: We start by converting u
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Unformatted text preview: 1 , u 2 , u 3 into an orthonormal basis, by letting v 1 = u 1 / || u 1 || = (1 / 2 , 1 / 2 , 1 / 2 , 1 / 2) T v 2 = u 2 / || u 2 || = (1 / 2 ,-1 / 2 , 1 / 2 ,-1 / 2) T v 3 = u 3 / || u 3 || = (1 / 2 , ,-1 / 2 , 0) T . Then, nd the projection p as p = h v 1 , x i v 1 + h v 2 , x i v 2 + h v 3 , x i v 3 = 5 v 1-v 2- 2 v 3 = ( 5 2 , 5 2 , 5 2 , 5 2 ) T-( 1 2 ,-1 2 , 1 2 ,-1 2 ) T-(1 , ,-1 , 0) T = (1 , 3 , 3 , 3) T ....
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