2.1 Vector spaces - Vector spaces V Definition 1 A real...

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Vector spaces Definition 1. A real vector space is a set of elements V together with two operations satisfying the following properties: α ¿ . If u and v are any elements of V , then u v is in V (i.e., V closed under the operation ) a). u v = v u for any u , v V ; b). u ( v w ) =( v u ) w for any u , v ,w V ; c). There is an element 0 in V such that u 0 = 0 u = u for any u V ; d). For each u in V , there is an element u in V such that u ( u ) = 0 ; β ¿ . If u is any element of V and c is real number, then c u is in V (i.e., V closed under the operation ) e). c ( u v ) = c u c v for all real numbers c and all u and v in V ; f). ( c + d ) u = c u d u; g). c ( d u ) =( cd ) u; h). 1 u = u.
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The elements of V are called vectors; the real numbers are called scalars. The operation is called vector addition; the operation is called scalar multiplication. The vector 0 in property c) is called a zero vector. The vector u in property d) is called a negative of u. Example 1. Consider the set R n = { a = ( α 1 2 ,…,α n ) : α i R } , where R is the set real numbers. If a = ( α 1 2 ,…,α n ) R n and b = ( β 1 , β 2 ,…, β n ) R n then we put a b = ( α 1 + β 1 2 + β 2 ,…,α n + β n ) and c a = ( 1 ,cα 2 ,…,cα n ) , c R.
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