1.1 Vector Spaces over a Commutative Field

 Let \( (K, +_K, \cdot_K, 0_K, 1_K) \) be a commutative field, where \( 0_K (resp. 1_K ) \)  denotes the identity element with respect to the internal operation\( +_K \) (resp. the internal operation \( \cdot_K\ ). \)

 Definition 1.1.1: Given a non-empty set \( E \), we say that \( (E, +_E, \cdot_E) \) is a vector space over the field \( K \) (or a \( K -vector space) if:  \)

1. \(  (E, +_E) \) is an abelian group.

2. \(  ._E \) is an external composition law (external operation).

\[\cdot_E: K \times E \longrightarrow E \] \[\alpha, x) \mapsto a \cdot_E x \] 

 satisfying (the four properties of the external operation) for all \( \alpha, \beta \in K \) and all \( x, y \in E \):

a. \(  \alpha \cdot_E (x +_E y) = ( \alpha \cdot_E x) +_E ( \alpha \cdot_E y) \).  

b. \( ( \alpha +_K \beta) \cdot_E x = ( \alpha \cdot_E x) +_E (\beta \cdot_E x) \). 

c. \( ( \alpha \cdot_K \beta) \cdot_E x =  \alpha \cdot_E ( \beta \cdot_E x) \). \)

d. \( 1_K \cdot_E x = x. \)

The elements of  The elements of E are called vectors, and those of \( K \) are called scalars. The internal composition law \( +_E \) is called the vector sum (vector addition), and the external composition law \( \cdot_E \) is called scalar multiplication.

Remark 1.1.1:

We denote by \( 0 = 0_E \) the identity element of \( E \) with respect to the operation \( +_E \), and we will write, when there is no ambiguity, \( + \) (resp. \( \cdot \)) instead of \( +_E \) (resp. \( \cdot_E \)), and therefore we will refer to the \( K \)-vector space \( (E, +, \cdot) \).