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115apracticeexam1-sol - Practice Hour Exam#1 Solutions Math...

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Practice Hour Exam #1 Solutions Math 115A Section 3 NAME: SOLUTIONS SCORES: 1. / 20 2. / 20 3. / 10 4. / 10 5. / 20 6. / 20 Total: / 100
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Problem 1 (20 points - 5 points each) (a) State what it means for a subset S of a vector space V to be linearly independent . S is linearly independent if whenever v 1 , . . . , v n S are distinct and a 1 , . . . , a n are scalars such that a 1 v 1 + · · · a n v n = ~ 0, then a 1 = · · · = a n = 0. (b) Define the dimension of a finite-dimensional vector space. The dimension of a finite-dimensional vector space V is the number of vectors in any basis for V . (c) Suppose T : V W is a linear transformation. Define the null space of T . The null space of T is the set N ( T ) = { x V | T ( x ) = ~ 0 W } . (d) Suppose T : V W is a linear transformation. Define the rank of T . The rank of T is the dimension of the range of T , dim( R ( T )).
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Problem 2 (20 points - 4 points each) For each of the following statements, determine if they are true or false. If they are true, prove them. If they are false, provide a counterexample . (a) dim( P n ( F )) = n . FALSE! Since { 1 , x, . . . , x n } is a basis for P n ( F ), we have dim( P n ( F )) = n + 1. (b) 3 distinct vectors in R 4 must be linearly independent. FALSE! There are many ways to disprove this. The fastest way is to say that if the set contains (0 , 0 , 0 , 0), then it must be linearly dependent. Another less perverse example is the set { (1 , 0 , 0 , 0) , (2 , 0 , 0 , 0) , (3 , 0 , 0 , 0) } .
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