sample_tt2_2_ans

sample_tt2_2_ans - Math 136 Sample Term Test 2 # 2 Answers...

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Math 136 Sample Term Test 2 # 2 Answers NOTE : - Only answers are provided here (and some proofs). On the test you must provide full and complete solutions to receive full marks. 1. Short Answer Problems a) Let S = { ~v 1 ,...,~v n } be a non-empty subset of a vector space V . Define the statement S is linearly independent. Solution: S is linearly independent if the only solution of c 1 ~v 1 + c 2 ~v 2 + ··· + c n ~v n = 0 is c 1 = c 2 = ··· = c n = 0. b) Write the definition of a subspace S of a vector space V . Solution: S is a subspace of V if S is a subset of V and S is a vector space using the same operations as V . c) Write the definition of the dimension of a vector space V . Solution: The dimension of V is the number of elements in any basis for V . d) Prove that 0 ~x = ~ 0 for any ~x V . Solution: By vector space axioms 3,4,2,8,10 and 4, 0 ~x = 0 ~x + ~ 0 = 0 ~x + ( ~x + ( - ~x )) = (0 ~x + ~x ) + ( - ~x ) = (0 + 1) ~x + ( - ~x ) = 1 ~x + ( - ~x ) = ~x + ( - ~x ) = ~ 0. e) Is it true that if a set S with more than one vector is linearly dependent then every vector ~v S can be written as a linear combination of the other vectors. Justify your answer. Solution: It is false.
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This note was uploaded on 11/25/2010 for the course MATH 136 taught by Professor All during the Spring '08 term at Waterloo.

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sample_tt2_2_ans - Math 136 Sample Term Test 2 # 2 Answers...

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