# Abstract Algebra/Vector Spaces

- 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