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Unformatted text preview: Math 136 Sample Term Test 2 # 1 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) What is the definition of the row space and column space of a matrix A . Solution: The row space is the spanning set of the rows of A and the column space is the spanning set of the columns of A . b) What is the definition of a subspace. Solution: A subspace is a subset of a vector space V that is also a vector space under the same operations as V . c) What is the definition of a basis. Solution: A linearly independent spanning set. d) Let B = { (1 , 2 , 1) , ( 1 , , 2) , (1 , 1 , 1) } . If [ ~v ] B = 1 1 1 what is ~v ? Solution: ~v = 1(1 , 2 , 1) + ( 1)( 1 , , 2) + (1)(1 , 1 , 1) = (3 , 3 , 0) . e) Let V be a vector space and ~v ∈ V . Prove that ( 1) ~v is the additive inverse of ~v . Solution: We have ~v + ( 1) ~v = (1 + ( 1)) ~v = 0 ~v = ~ , by V8. Thus ( 1) ~v is the additive inverse. 2. Let β = { 1 + x 2 , 1 + x + x 2 , 1 + 2 x + 2 x 2 } . a) Show that β is a basis for P 2 . Solution: Consider c 1 (1 + x 2 ) + c 2 (1 + x + x 2 ) + c 3 (1 + 2 x + 2 x 2 ) = 0. Rowreducing the coefficient matrix gives 1 1 1 0 1 2 1 1 2 ∼ 1 1 1 0 1 2 0 0 1...
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 Spring '08
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 Math, Linear Algebra

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