It is time to consider the idea of a Vector space.
Definition 7.1.1 A vector space is an Abelian group of “vectors” satisfying the axioms of an Abelian group,

the commutative law of addition,

the associative law for addition,

the existence of an additive identity,

the existence of an additive inverse, along with a field of “scalars”, F which are allowed to multiply the vectors according to the following rules. (The Greek letters denote scalars.)
 (7.1) 
 (7.2) 
 (7.3) 
 (7.4) 
The field of scalars is usually ℝ or ℂ and the vector space will be called real or complex depending on whether the field is ℝ or ℂ. However, other fields are also possible. For example, one could use the field of rational numbers or even the field of the integers mod p for p a prime. A vector space is also called a linear space.
For example, ℝ^{n} with the usual conventions is an example of a real vector space and ℂ^{n} is an example of a complex vector space. Up to now, the discussion has been for ℝ^{n} or ℂ^{n} and all that is taking place is an increase in generality and abstraction.
There are many examples of vector spaces.
Example 7.1.2 Let Ω be a nonempty set and let V consist of all functions defined on Ω which have values in some field F. The vector operations are defined as follows.
Note that F^{n} actually fits in to this framework. You consider the set Ω to be
Example 7.1.3 Generalize the above example by letting V denote all functions defined on Ω which have values in a vector space W which has field of scalars F. The definitions of scalar multiplication and vector addition are identical to those of the above example.