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115a-4exam1sol - MATH 115A Lecture 4 Fall 2008 Midterm 1...

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MATH 115A - Lecture 4 - Fall 2008 Midterm 1 - October 20, 2008 NAME: STUDENT ID #: This is a closed-book and closed-note examination. Calculators are not allowed. Please show all your work. Use only the paper provided. You may write on the back if you need more space, but clearly indicate this on the front. There are 5 problems for a total of 100 points. POINTS: 1. 2. 3. 4. 5. 1
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2 1. (10 points each) For the given subset W of the F -vector space V , prove or disprove that W is a subspace of V . (a) F = R , V = R 3 and W = { ( a 1 ,a 2 ,a 3 ) V | a 1 - a 2 = a 3 } . Clearly, the zero vector (0 , 0 , 0) is in W . Now suppose a = ( a 1 ,a 2 ,a 3 ) and b = ( b 1 ,b 2 ,b 3 ) are in W . Then ( a 1 + b 1 ) - ( a 2 + b 2 ) = a 1 - a 2 + b 1 - b 2 = a 3 + b 3 , so a + b W . If λ R is a scalar, then λa 1 - λa 2 = λ ( a 1 - a 2 ) = λa 3 , so λ a W . That is, W is closed under addition and scalar multiplication and contains 0, hence, it is a subspace. (b)
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115a-4exam1sol - MATH 115A Lecture 4 Fall 2008 Midterm 1...

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