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mat67-Homework8

# mat67-Homework8 - MAT067 University of California Davis...

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MAT067 University of California, Davis Winter 2007 Homework Set 8: Exercises on Inner Product Spaces Directions : Please work on all of the following exercises and then submit your solutions to the Calculational Problems 1 and 8, and the Proof-Writing Problems 2 and 11 at the beginning of lecture on March 2, 2007. 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 would you obtain if you applied the Gram-Schmidt process to the basis ( f 3 , f 2 , f 1 )? 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 . 3. Let n Z + be a positive integer, and let a 1 , . . . , a n , b 1 , . . . , b n R be any collection of 2 n real numbers. Prove that n k =1 a k b k 2

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