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.