602 - Example: an orthogonal set By definition, a set with...

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1 Example: an orthogonal set By definition, a set with only one vector is an orthogonal set. Proof Let S = { v 1 , v 2 , , v k } R n be an orthogonal set and v i 0 for i = 1, 2, , k . If c 1 , c 2 , , c k make c 1 v 1 + c 2 v 2 + " + c k v k = 0 ,then for i = 1, 2, , k , we have 0 , i.e., c i = 0. Definition. A basis that is an orthogonal set is called an orthogonal basis. Example: The standard basis E of R n is an orthogonal basis. u u u u u
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2 Example: S = { v 1 , v 2 , v 3 } is an orthogonal basis for R 3 with Proof u u u and u u u u u Proof By induction on k . The theorem obviously holds for k = 1. Assume the theorem holds for k 1, and consider the case for k + 1. We have 1. In the set { v 1 , v 2 , , v k , v k +1 }, v 1 , v 2 , , v k are nonzero orthogonal vectors, and Span{ v 1 , v 2 , , v k } = Span{ u 1 , u 2 , , u k }. 2. v k +1 v i = 0 for i = 1, 2, , k , since
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3 . 2 1 1 2 1 1 1 2 1 1 2 1 1 1 1 2 1 1 1 1 1 i k k k k i i i i k i i i i k i i i i k i k i k i k v v v v u v v v v u v v v v u v v v v u v v v v u v u v v + + + + + + + + + +
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This note was uploaded on 10/16/2010 for the course EE 155 taught by Professor Fong during the Spring '09 term at National Taiwan University.

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602 - Example: an orthogonal set By definition, a set with...

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