Abstract Algebra/Vector Spaces

< Abstract Algebra

DefinitionEdit

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  
  4.  
  5.  

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.  
  2.  
  3.  
Proofs:
  1.  
  2. We want to show that  , but  
  3. Suppose   such that  , then