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Unformatted text preview: MA 265 LECTURE NOTES: MONDAY, FEBRUARY 18 Real Vector Spaces Properties of nSpace. In the previous lecture, we studied properties of V = ( R 2 as 2space; and R 3 as 3space. If u , v , w V and c, d R then the following properties are valid: ( Commutativity: ) u + v = v + u . ( Associativity :) u + ( v + w ) = ( u + v ) + w , and c ( d u ) = ( cd ) u . ( Distributivity: ) c ( u + v ) = c u + c v , and ( c + d ) u = c u + d u . ( Identity: ) There exists V such that u + = u . Moreover, 1 u = u . ( Inverses: ) There exists u V such that u + ( u ) = . We will prove these statements later in the lecture. Vector Spaces Definition. We generalize the ideas above. A real vector space is a triple ( V, , ) such that if u , v , w V and c, d R then the following properties are valid: ( Commutativity: ) u v = v u . ( Associativity :) u ( v w ) = ( u v ) w , and c ( d u ) = ( cd ) u ( Distributivity: ) c ( u v ) = ( c u ) ( c v ), and ( c + d ) u = ( c u ) ( d u ). ( Identity: ) There exists V , called the zero vector , such that u = u . Moreover 1 u = u ( Inverses: ) There exists u V , called the negative of u , such that u + ( u ) = . The elements u V are called vectors and the elements c R are called scalars . The operation : V V V which sends ( u , v ) 7 u v is called vector addition . The operation : R V V which sends...
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 Spring '10
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 Addition, Vector Space, real vector space

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