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 A as a mapping of A \times A to A. This may be referred to as an internal composition in A. Now, let A and B be two non-empty sets. Then a mapping f:A \times B \to B is called an external composition in B over A.

Definition: Let \left( {F, + , \times } \right) be a field. Then a set V is called a vector space over the field F if V is an abelian group under an operation which is denoted by  + , and if for every a \in F, u \in V there is defined an element au in V such that

(i) a\left( {u + v} \right) = au + av, for all a \in F, u,v \in V.

(ii) \left( {a + b} \right)u = au + bu, for all a,b \in F, u \in V.

(iii) a\left( {bu} \right) = \left( {ab} \right)u, for all a,b \in F, u \in V.

(iv) 1 \cdot u = u \cdot 1 represents the unity element of F under multiplication.

The following notations will be constantly used in the forthcoming tutorials.

(1) Generally F will be the field whose elements shall often be referred to as scalars.

(2) V will denote the vector space over F whose elements shall be called vectors.

Thus to test that V is a vector space over F, the following axioms should be satisfied:

(V1): \left( {V, + } \right) is an abelian group.

(V2): Scalar multiplication is distributive over addition in V, i.e. a\left( {u + v} \right) = au + av, for all a \in F, u,v \in V.

(V3): Distributive of scalar multiplication over addition in F, i.e. \left( {a + b} \right)u = au + bu, for all a,b \in F, u \in V.

(V4): Scalar multiplication is associative, i.e. a\left( {bu} \right) = \left( {ab} \right)u, for all a,b \in F, u \in V.

(V5): Property of unity: Let 1 \in F be the unity of F, then 1 \cdot u = u \cdot 1 for all u \in V.

A vector space V over a field F is expressed by writing V\left( F \right). Sometimes writing only V is sufficient provided the context makes it clear which field has been considered.

If the field is \mathbb{R}, the set of real numbers, then V is said to be a real vector space. If the field is \mathbb{Q}, the set of rational numbers, then V  is said to be a rational vector space. Finally, if the field is \mathbb{C}, the set of complex numbers, V is called a complex vector space.