Abstract algebra II - Essay Example

As a start, consider that the set of scalars given above could be generalized by letting the scalars be elements of a field. Also, the first five axioms all involve addition and hence constitute a special algebraic object. Lastly, the closure item (#6 in the above list) can be included as part of the definition of scalar multiplication in the introductory statement. Given a positive integer n (the dimension) and a field of scalars F, a vector space over the field F is V = Fn. This vector space has elements that are n-tuples: u = (u1, u2, ., un). The components ui are in the field of scalars F.

Scalar multiplication can then be defined as the map F x V V, denoted as c u, where c is an element of the field F: so that multiplication of a vector u in the vector space V by a scalar c is equivalent to multiplication of each of the scalar components ui by the scalar c to form a new n-tuple, or vector within the vector space. The modern definition of a vector space V over a field F can then be defined as consisting of a set of elements (vectors), such that additive closure (axiom #1) and closure under scalar multiplication (axiom #6) are part of the definition, and not needed as separate axioms.

By definition, this vector space V over a field F is an Abelian group under addition. The axioms 2-5 state that V is an Abelian group; these separate axioms are unnecessary if this property is used in the definition of the vector space. This leaves axioms 7-10. This statement in this axiom can be reduced to a distributive law of fields holding component by component. This can be seen by using the definitions of vector addition and scalar multiplication, and the definition of a vector in terms of its components: Using the definition of scalar multiplication and the definition of a vector in terms of its components, this axiom can be reduced to a distributive law for fields, holding component by component, between the scalars of a field: This