m235notes15 - Notes 15 Vector Spaces Lecture November 2011...

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Notes 15: Vector Spaces Lecture November, 2011 Definition 1. A vector space is a set V with an operation we call addition and a map that associates to an element λ R and an element v V another element in V denoted by λv. We denote the addition of two elements v, w in V by v + w. These operations satisfy v+w=w+v. (u+v)+w=u+(v+w). There exists an element of V, which we denote by 0 , so that 0 + v = v + 0 = v for all v V. For all elements v V there exists an element ( - v ) V so that v + ( - v ) = 0 . 1 · v = v for all v V. a ( bv ) = ( ab ) v for all a, b R , v V . 0 · v = 0 for all v V . Here the first 0 is the zero in R and the second is the additive identity in V . ( a + b ) v = av + bv for all a, b R , v V . a ( v + w ) = av + aw for all a R , v, w V. Example 1 . R n is a vector space. Example 2 . Let R n × m denote the set of all n × m matrices. Example 3 . Let P n denote the set of all polynomial functions of degree less than or equal to n in one variable. We define addition as follows: Let f, g
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