assign3_08

assign3_08 - The Chinese University of Hong Kong Department...

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The Chinese University of Hong Kong Department of Systems Engineering & Engineering Management ERG 2018 Advanced Engineering Mathematics (2008) Assignment 3 Question 1 The Gram-Schmidt (GS) orthogonalization procedure extends the idea of orthogonal projection (see Assignment 2 Q3(a)), to obtain a set of k or- thonormal vectors { ~e 1 ,~e 2 , ..., ~e k } from k linearly independent vectors { ~u 1 ,~u 2 , ..., ~u k } . Denote proj q ~p as the orthogonal projection of ~p on ~q . The k sequential steps of GS can be described as follows: Normalize 1 to obtain 1 Normalize 2 proj e 1 2 to obtain 2 ... Normalize k k - 1 j =1 proj e j k to obtain k (a) Find 1 2 3 for 1 =[1 , 1 / 2 , 1 / 3] T , 2 / 2 , 1 / 3 , 1 / 4] T ,and 3 = [1 / 3 , 1 / 4 , 1 / 5] T . Discuss any observations in the normalization of the vectors. Note that you should always verify the LI requirement before applying GS. (b) Prove the GS procedure using mathematical induction. Give reasons on why normalization can be carried out in each step as this would presume a non-zero vector norm under all situations.

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