- Definition (Vector Space)
- Let F be a field. A set V with two binary operations: + (addition) and
(scalar multiplication), is called a Vector Space if it has the following properties:
forms an abelian group
for
and 
for
and 


The scalar multiplication is formally defined by
, where
.
Elements in F are called scalars, while elements in V are called vectors.
- Some Properties of Vector Spaces



- Proofs:

- We want to show that
, but 
- Suppose
such that
, then 