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Unformatted text preview: 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 , , 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 , , 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...
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This note was uploaded on 03/15/2012 for the course MATH 115A 262398211 taught by Professor Fuckhead during the Winter '10 term at UCLA.
- Winter '10