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Unformatted text preview: 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 R and an element v V another element in V denoted by v. 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 . 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....
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