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mat67-Homework8_Solutions

mat67-Homework8_Solutions - MAT067 University of California...

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MAT067 University of California, Davis Winter 2007 Solutions to Homework Set 8 As usual, we are using F to denote either R or C . We also use · , · to denote an arbitrary inner product and · to denote its associated norm. 1. Let ( e 1 , e 2 , e 3 ) be the canonical basis of R 3 , and define f 1 = e 1 + e 2 + e 3 f 2 = e 2 + e 3 f 3 = e 3 . (a) Apply the Gram-Schmidt process to the basis ( f 1 , f 2 , f 3 ). (b) What do you obtain if you applied the Gram-Schmidt process to the basis ( f 3 , f 2 , f 1 )? Solution : (a) Given the vectors f 1 = (1 , 1 , 1) , f 2 = (0 , 1 , 1) , f 3 = (0 , 0 , 1) R 3 , we apply the Gram-Schmidt procedure to form the orthonormal basis { u 1 , u 2 , u 3 } for R 3 as follows: u 1 = f 1 f 1 = (1 , 1 , 1) (1) 2 + (1) 2 + (1) 2 = 1 3 , 1 3 , 1 3 , u 2 = f 2 f 2 , u 1 u 1 f 2 f 2 , u 1 u 1 = f 2 2 3 u 1 f 2 2 3 u 1 = ( 2 3 , 1 3 , 1 3 ) ( 2 3 ) 2 + ( 1 3 ) 2 + ( 1 3 ) 2 = 2 6 , 1 6 , 1 6 , u 3 = f 3 f 3 , u 1 u 1 f 3 , u 2 u 2 f 3 f 3 , u 1 u 1 f 3 , u 2 u 2 = · · · = ( 0 , 1 2 , 1 2 ) (0) 2 + ( 1 2 ) 2 + ( 1 2 ) 2 = 0 , 1 2 , 1 2 .
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Solution : (b) The identical technique can be applied to this part. Applying the Gram-Schmidt procedure, we form the orthonormal basis { v 1 , v 2 , v 3 } for R 3 as follows: v 1 = f 3 f 3 = (0 , 0 , 1) = e 3 , v 2 = f 2 f 2 , u 1 u 1 f 2 f 2 , u 1 u 1 = f 2 u 1 f 2 u 1 = (0 , 1 , 0) 0 2 + 1 2 + 0 2 = (0 , 1 , 0) = e 2 , v 3 = f 1 f 1 , u 1 u 1 f 1 , u 2 u 2 f 1 f 1 , u 1 u 1 f 1 , u 2 u 2 = f 1 u 1 u 2 f 1 u 1 u 2 = (1 , 0 , 0) 1 2 + 0 2 + 0 2 = (1 , 0 , 0) = e 1 . However, this should be clear from the fact that we have started by setting v 1 = e 3 , then v 2 = e 2 + e 3 e 3 , and finally v 3 = e 1 + e 2 + e 3 e 3 e 2 . 2
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2. Let V be a finite-dimensional inner product space over F . Given any vectors u, v V , prove that the following two statements are equivalent: (a) u, v = 0 (b) u u + αv for every α F . Solution : Proposition. Let V be a finite-dimensional inner product space over F . Given any vectors u, v V , u, v = 0 if and only if u u + αv for every α F . Proof. Let u, v V be any two vectors in V . (= ) Suppose that u, v = 0, and let α F . Then u, αv = α u, v = α 0 = 0 so that u and αv are orthogonal vectors in V . Thus, using that αv 2 0 and applying the Pythagorean Theorem (Theorem 6.3 in the textbook) to u and αv , we obtain u 2 u 2 + αv 2 = u + αv 2 .
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