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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.

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