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 uvw =+ where The vector v is parallel to a given nonzero vector q . The vector w is orthogonal to q . q v pro ju = vv = q q    We have cos q vu q q = Recall the definition of dot product, we can get the form uq v q = g q q () ue = g e , where q e q = 1.1 Example: Let 4 2 3 u   =-  and 3 0 1 q = . Express u in the form where v is parallel to q and w is orthogonal to q . 2. Gram-Schmidt process Given a set of n LI vectors 12 { ,, ,} n V v = v K , Gram-Schmidt process generate a set of n ortho-normal vectors { n E e ee = v K , such that Span V = Span E . It works as follows: 11 uv = , 1 1 1 u e u = 22 211 ( ), vee g 2 2 2 u e u = 33 31 1 322 ( ) ve e = -- gg 3 3 3 u e u = M M 1 1 n n n njj j u v - = g n n n u e u =
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Given a set of n vectors 12 { ,, ,} n V v vv = v K , your task is to obtain a set of k ortho-normal vectors { k E e ee = v K , such that Span V = SpanE . (1) It is obvious that kn , what does it mean when the strict inequality holds, i.e. < ? (2) When < , you can either delete some vectors in V to form a linearly independent subset of vectors, and then find the corresponding ortho-normal vectors, OR you can do Gram-Schmidt process immediately without deletion, what will happen during the procedure? 2.1 Example: Let 1011 1,2 , 1,1 0 1 16 V         =-  -     , find E . Pay attention to the above questions during the procedure.
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This note was uploaded on 11/13/2010 for the course SEEM SEEM2018 taught by Professor Chan during the Spring '10 term at CUHK.

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

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