20F Quiz 3 - Name TA Math 20F Quiz 3 May 9 2008 1 Let V be...

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Name: PID: TA: Sec. No: Sec. Time: Math 20F Quiz 3 May 9, 2008 1. Let V be the set of all vectors in R 3 of the form - a + 1 a - 6 b 2 b + a , where a and b are real numbers. Is V a vector space? If so, find a basis of V . Otherwise, explain why V is not a vector space. Solution: V is not a vector space because V does not contain 0 = 0 0 0 . To see this, note that the first entry is zero if and only if a = 1. For the second entry to be zero, we need 1 - 6 b = 0, so b = 1 6 . But the third entry is 2 b + a = 1 3 + 1, which is not zero. 2. Mark each of the following as true or false. a. For an m × n matrix A , Nul A is in R m . False. Nul A is a subspace of R n , not R m . b. Col A is the set of all vectors that can be written as A x for some x . True. c. If H = Span { v 1 ,...,v k } then a subset of { v 1 ,...,v k } forms a basis of H . True. This is the content of Theorem 5 on page 239, as long as not all the vectors v 1 ,...,v k are zero. If all these vectors are zero, H must be the zero vector space { 0 } . In this case, it is still true that a subset of { v 1 ,...,v k }
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