solns5 - MATH 235/W08: Orthogonality & Gram-Schmidt,...

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Unformatted text preview: MATH 235/W08: Orthogonality & Gram-Schmidt, SOLUTIONS to Assignment 5 Solutions to questions 3,9,11,13,14. Contents 1 Orthogonal Pairs 2 1.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Representation using Orthogonal Complement 2 2.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 Basis and Distance *** 3 3.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.1.1 A Helpful MATLAB Program with Output . . . . . . . . . . . . . . . . . . . 4 4 Orthogonal Basis 5 5 Orthogonal Basis for Columns Space 5 6 Orthogonal Matrix 6 7 Orthogonal Matrix Two 6 8 True-False Questions 6 9 Angles Between Vectors *** 6 9.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 10 Orthonormal Bases 7 11 Basis and Nearest Vectors *** 7 11.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 12 Gram-Schmidt Procedure 8 13 Gram-Schmidt Procedure and Inner Products *** 8 13.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 14 MATLAB: Predator-Prey System *** 10 14.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1 1 Orthogonal Pairs Determine which pairs of vectors are orthogonal; find the norms of each vector. a) u = i 2 1 , v = i 2 1 b) u = 1 / 2 1 / 2 1 / 2 1 / 2 , v = 1 / 2 2 1.1 Solution BEGIN SOLUTION: a) ( u,v ) = u T v = i 2 1 T i 2 1 = 1 + 2 1 negationslash = 0 . Therefore, these vectors are not orthogonal. The norms of the two vectors are the same bardbl u bardbl = bardbl v bardbl = 1 + 2 + 1 = 2. b) ( u,v ) = u T v = 1 / 2 1 / 2 1 / 2 1 / 2 T 1 / 2 2 = 1 2 2 2 2 negationslash = 0 . Again, this pair of vectors is not orthogonal. The norms are bardbl u bardbl 2 = 4 / 4 = 1 and bardbl v bardbl 2 = 1 2 2 = 3 2 . END SOLUTION. 2 Representation using Orthogonal Complement Let u 1 = (3 , , 1) T , u 2 = (0 , 1 , 0) T , and y = (0 , 3 , 10) T . Let W = span { u 1 , u 2 } . Represent y as the sum of a vector in W and a vector in the orthogonal complement of W . 2.1 Solution BEGIN SOLUTION: Note that u 1 ,u 2 are orthogonal. Therefore, the projection onto W is easily found to be P W y = 10 10 u 1 + 3 1 u 2 = u 1 + 3 u 2 = 3 3 1 . 2 And y = P w y + ( y P W y ) = 3 3 1 + 3 9 is the required representation....
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This note was uploaded on 04/27/2010 for the course MATH 235 taught by Professor Celmin during the Winter '08 term at Waterloo.

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solns5 - MATH 235/W08: Orthogonality & Gram-Schmidt,...

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