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T5 - ERG 2018 Tutorial 5 October 9 2008 1 Projection...

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ERG 2018 Tutorial 5 October 9, 2008 1. Projection, Orthogonal Vectors We can express a given vector u in the form u v w = + where The vector v is parallel to a given nonzero vector q . The vector w is orthogonal to q . q v proj u = v v = q q We have cos q v u q q = Recall the definition of dot product, we can get the form u q v q = g q q ( ) u e = g e , where q e q = 1.1 Example: Let 4 2 3 u = - and 3 0 1 q     =       . Express u in the form u v w = + where v is parallel to q and w is orthogonal to q . 2. Gram-Schmidt process Given a set of n LI vectors 1 2 { , , , } n V v v v = v v v K , Gram-Schmidt process generate a set of n ortho-normal vectors 1 2 { , , , } n E e e e = v v v K , such that Span V = Span E . It works as follows: 1 1 u v = , 1 1 1 u e u = 2 2 2 1 1 ( ) , u v v e e = - g 2 2 2 u e u = 3 3 3 1 1 3 2 2 ( ) ( ) u v v e e v e e = - - g g 3 3 3 u e u = M M 1 1 ( ) n n n n j j j u v v e e - = = - g n n n u e u =
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