115as11exam1-sol

# 115as11exam1-sol - Hour Exam#1 Math 115A Section 3 NAME...

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Unformatted text preview: Hour Exam #1 Math 115A Section 3 April 18, 2011 NAME: SOLUTIONS Bruin ID: SCORES: 1. / 20 2. / 20 3. / 20 4. / 20 5. / 20 Total: / 100 SIGNATURE: I hereby swear that in taking this exam I have adhered to all of the university’s rules on academic integrity. Problem 1 (20 points - 5 points each) (a) State what it means for a subset S of a vector space V to span V . S spans V if, for any x ∈ V , there are x 1 ,...,x n ∈ S and a 1 ,...,a n ∈ F such that x = a 1 x 1 + ··· + a n x n . (b) State what it means for a vector space to be infinite-dimensional . V is infinite-dimensional if V is not finite-dimensional, that is, V does not have a finite spanning set. (c) Suppose T : V → W is a linear transformation. Define the range of T . The range of T is R ( T ) = { y ∈ W : there is x ∈ V such that T ( x ) = y } . (d) Suppose T : V → W is a linear transformation. Define the nullity of T . The nullity of T is the dimension of the null space of T . 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( M m × n ( F )) = mn . This is true: the set { E ij : 1 ≤ i ≤ m, 1 ≤ j ≤ n } is a basis for M m × n ( F ), where E ij is the matrix with a 1 in the ( i,j )-entry and 0 in all other entries.)-entry and 0 in all other entries....
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115as11exam1-sol - Hour Exam#1 Math 115A Section 3 NAME...

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