Abstract Algebra/Vector Spaces

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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:
  1. forms an abelian group
  2. for and
  3. for and

The scalar multiplication is formerly defined by , where .

Elements in F are called scalars, while elements in V are called vectors.

Some Properties of Vector Spaces
  1. We want to show that , but
  2. Suppose such that , then