# Vector Space

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.