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Unformatted text preview: Week 9 Chapter 6: Vector Spaces We have already looked at a few number systems in this course that have displayed similar algebraic properties. For example, R , matrices, vectors in R 2 , R 3 ,..., R n . In terms of linear operations, they have all shared properties such as associativity, distributivity, commutative addition, a zero element, and an identity element. We can deal with these number systems and many others simultaneously by introducing the notion of an abstract vector space. Definition: A vector space V is a non-empty set of objects called vectors with two operations called addition and scalar multiplication subject to the following 10 axioms below. ( These terms are in quotes because we will refer to the objects as vectors but they may actually be matrices or polynomials or real numbers etc. The addition and scalar multiplication are in quotes since these operations will be have varying definitions depending on the object that the operation is applied to. ) Addition Axioms: A1. If ~u and ~v are in V ,then ~u + ~v V . A2. ~u + ~v = ~v + ~u A3. ~u + ( ~v + ~w = ( ~u + ~v ) + ~w A4. There is a ~ V such that ~v + ~ 0 = ~v for all ~v V . A5. There is a- ~v V for each ~v V such that ~v + (- ~v ) = ~ 0. Scalar Multiplication Axioms: S1. If ~v V , then a~v V for all a R . S2. a ( ~v + ~w ) = a~v + a~w for all ~v, ~w V and all a R S3. ( a + b ) ~v = a~v + b~v for all ~v V,a,b R . S4. a ( b~v ) = ( ab ) ~v for all ~v V,a,b R S5. 1 ~v = ~v for all ~v V Examples: 1. R is a vector space where the vectors are real numbers and addition and scalar multiplication are the operations you learned long, long ago....
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