This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: VECTORS SPACES OVER R JOS E MALAG ON-L OPEZ The algebraic structure that we saw in R n : addition, scalar multipli- cation and the way these operations are compatible, is not exclusive of R n . The notion of vector space is the abstraction of such structure, and consists of the minimum properties required in order to work with such objects. Before we see other algebraic structures sharing similar properties we will give the formal definition of a vector space over R . Formal Definition of a Vector Space Over R n In order to define a vector space we require two sets: a collection of numbers that will play the role of scalars, and the collection of vec- tors. In this course, the scalars will be the real numbers. With this, the formal definition of a vector space (over R ) is as follows: A vector space over R is a set V endowed with two operations: addition +, and scalar multiplication , such that (1) V is closed under addition . For all v and w in V , there is a unique v + w in V . (2) Commutativity . For all v and w in V , v + w = w + v . (3) Associativity . For all u , v and w in V , ( u + v ) + w = u + ( v + w ). (4) Existence of Additive Identity . There is an element in V such that for all v in V , v + = v . (5) Existence of Additive Inverse . For all v in V there is an element v such that v + v = . (6) V is closed under scalar multiplication . For all v in V and all in R , there is a unique v in V . (7) For all v in V , 1 v = v . (8) For all v in V , and all and in R , ( ) v = ( v )....
View Full Document