# 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.