Fall 2010 Quiz Solutions

Fall 2010 Quiz Solutions - Math 140A, Fall 2010, Quiz,...

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Math 140A, Fall 2010, Quiz, 10/15/10 Instructions . Answer all questions. You may use without proof anything which was proved in class. If you need to cite a theorem, do so either by name, if it has one, or by brieFy stating what it says. 1. (10 points) Let a, b be positive real numbers, p and q non-zero positive integers. Explain what it means for b 1 /p = a . Using this, show that ( b 1 /p ) 1 /q = b 1 /pq . b 1 /p = a if a is the unique positive number such that a p = b . Note that ( a q ) p = a pq since p, q are integers, for a a positive real number. If a = b 1 /pq , then b = a pq = ( a q ) p . Thus a q = b 1 /p , so a = ( b 1 /p ) 1 /q . 2. (10 points) Show that if x R k and x · y = 0 for all y R k , then x = 0 . Suppose x n = 0, say x = ( x 1 , . . ., x k ) with some x i n = 0. Then let y be the vector all of whose components are zero except the
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