Vector Spaces

Definition:

• Set of vectors .
• Two operations. For :

  Addition:

  .

  Scalar multiplication:

  , where is a member of some field , (i.e. ).

Axioms:

Addition Commutes:

Addition Associates:

Scalar Multiplication Distributes:

Unique Zero Element:

Field Unit Element:

Span:

• Suppose .
• spans iff any can be written as .
• This is written .
• is a linear combination of the vectors in .

Basis:

• Any minimal spanning set is a basis.
• All bases are the same size.

Dimension:

• The number of vectors in any basis.
• We will work in 2 and 3 dimensional spaces.

Note:

In the definition of a Vector Space:

• No notion of distance or size
• No notion of angles
• No ``points''

These are Euclidean and Affine space concepts.