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Unformatted text preview: MA 265 LECTURE NOTES: MONDAY, MARCH 31 GramSchmidt Process Orthogonalization. Let ( V, + , Â· , ( , ) ) be an inner product space. Say that T * = { v 1 , v 2 , ..., v n } is a collection of of nonzero vectors, and let W = span T * be their span. Last time, we showed that there is a spanning set T = { w 1 , w 2 , ..., w n } consisting of normal vectors. Indeed, we defined the vectors w i = 1  v i  v i for i = 1 , 2 , ..., n. Say that S = { u 1 , u 2 , ..., u n } is a collection of of nonzero vectors, and let W = span S be their span. Today, we show that there is a spanning set T * consisting of orthogonal vectors. This will be an inductive process, which depends on the size n of S . First say that n = 1 i.e. S = { u 1 } . Then there is nothing to check because one cannot take the inner product of a pair of vectors when there is just one , so we may choose T * = { u 1 } . Now say that n = 2 i.e. S = { u 1 , u 2 } . We will choose T * = { v 1 , v 2 } in terms of v 1 = u 1 v 2 = u 2 + a 21 v 1 for some scalar c âˆˆ R . It is easy to see that W = span T * . We want T * to be orthogonal, so we would like 0 = ( v 1 , v 2 ) = ( v 1 , u 2 + a 21 v 1 ) = ( v 1 , u 2 ) + a 21 ( v 1 , v 1 ) = â‡’ a 21 = ( u 2 , v 1 ) (...
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This note was uploaded on 03/11/2010 for the course MA 261A 0026100 taught by Professor ... during the Spring '10 term at Purdue University Calumet.
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