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10w_math115a_hw2_solutions

10w_math115a_hw2_solutions - Math 115A Homework 2 Solutions...

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Math 115A Homework # 2 Solutions Prof. Yehuda Shalom TA: Darren Creutz Date Due: 21 Jan 2010 1.4.1 (a) T: write 0 = 0 x for any x in the set (b) F: 0 is in span ( ) (c) T: Thm 1.5 implies that span ( S ) W for any subspace W containing S so span ( S ) is contained in the intersection of all such subspaces; span ( S ) itself is a subspace containing S so the intersection of all such subspaces is contained in span ( S ). (d) F: any nonzero constant is ok (e) T: the new equation will contain the information from the old one (f) F: 0 = 1 has no solutions 1.4.6 Show that the vectors (1 , 1 , 0), (1 , 0 , 1) and (0 , 1 , 1) generate F 3 . Proof. Let v F 3 . Then v = ( a, b, c ) for some a, b, c F . Set r = 1 2 ( a + b - c ) s = 1 2 ( a - b + c ) t = 1 2 ( - a + b + c ) Then r (1 , 1 , 0) + s (1 , 0 , 1) + t (0 , 1 , 1) = ( r + s, r + t, s + t ) = 1 2 ( a + b - c + a - b + c, a + b - c - a + b + c, a - b + c - = ( a, b, c ) making use of the axioms of vector spaces (commutativity and definition of scalar multiplication). Hence v is a linear combination of the given three vectors. So span ( { (1 , 1 , 0) , (1 , 0 , 1) , (0 , 1 , 1) } ) = F 3 1.4.12 Show that a subset W of a vector space V is a subspace if and only if W =
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