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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 = braceleftBigg 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 ) mapsto→ u ⊕ v is called vector addition . The operation ⊙ : R × V → V which sends ( c, u ) mapsto→ c ⊙ u is called scalar multiplication . Whenever the properties are valid for c, d ∈ C , we call ( V, ⊕ ,, ⊙ ) a complex vector space . In either case, we sometimes abuse notation and say “ V is a real vector space” instead of saying “( V, ⊕ , ⊙ ) is a real vector space”....
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This note was uploaded on 02/25/2010 for the course MA 00265 taught by Professor ... during the Spring '10 term at Purdue University Calumet.
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