Proposition 1.1.1: Let \( E \) be a \( K \)-vector space, we have:

1. \( \forall x \in E, \ 0_K \cdot x= 0_E \).
2. \( \forall \alpha \in K, \ \alpha \cdot 0_E = 0_E \)
3. \( \forall \alpha \in K, \forall x \in E, \ (-\alpha) \cdot x= -(\alpha\cdot x) = \alpha \cdot (-x) \)  
4. \( \forall \alpha \in K, \ \forall x \in E, \ \alpha \cdot x= 0_E \iff \alpha= 0_K \text{ or } x= 0_E \). 
5. \( \forall  \alpha \in K \) \( \forall x \in E \), \(\alpha - \beta) \cdot x = \alpha \cdot x - \beta \cdot x \) 
6. \( \forall  \alpha \in K \) \( \forall x,y \in E , \alpha \cdot (x - y)  = \alpha \cdot x - \alpha \cdot y\ \)

Example 1.1.1:

1.The trivial vector space is the vector space over a field \( K \) that contains only a single element, which must necessarily be the zero vector. \item Any field \( K \) can be viewed as a \( K \)-vector space. The addition and multiplication of \( K \) provide the vector sum and scalar multiplication, respectively.

 2. More generally, the set of \( n \)-tuples of elements from \( K \), equipped with the usual operations, forms the vector space \( K^n \).