Before giving the formal definition of an abstract vector space, we define what is known as an external composition in one set over another. We have already defined a binary composition in a set as a mapping of to . This may be referred to as an internal composition in . Now, let and be two non-empty sets. Then a mapping is called an external composition in over .
Definition: Let be a field. Then a set is called a vector space over the field if is an abelian group under an operation which is denoted by , and if for every , there is defined an element in such that
(i) , for all , .
(ii) , for all , .
(iii) , for all , .
(iv) represents the unity element of under multiplication.
The following notations will be constantly used in the forthcoming tutorials.
(1) Generally will be the field whose elements shall often be referred to as scalars.
(2) will denote the vector space over whose elements shall be called vectors.
Thus to test that is a vector space over , the following axioms should be satisfied:
(V1): is an abelian group.
(V2): Scalar multiplication is distributive over addition in , i.e. , for all , .
(V3): Distributive of scalar multiplication over addition in , i.e. , for all , .
(V4): Scalar multiplication is associative, i.e. , for all , .
(V5): Property of unity: Let be the unity of , then for all .
A vector space over a field is expressed by writing . Sometimes writing only is sufficient provided the context makes it clear which field has been considered.
If the field is , the set of real numbers, then is said to be a real vector space. If the field is , the set of rational numbers, then is said to be a rational vector space. Finally, if the field is , the set of complex numbers, is called a complex vector space.