A vector space over is a set of Vectors for which any Vectors , , and and any Scalars , have the following properties:

- 1. Commutativity:

- 2. Associativity of vector addition:

- 3. Additive identity: For all ,

- 4. Existence of additive inverse: For any , there exists a such that

- 5. Associativity of scalar multiplication:

- 6. Distributivity of scalar sums:

- 7. Distributivity of vector sums:

- 8. Scalar multiplication identity:

distinct Subspaces of Dimension .

A Module is abstractly similar to a vector space, but it uses a Ring to define Coefficients instead of the Field used for vector spaces. Modules have Coefficients in much more general algebraic objects.

**References**

Arfken, G. *Mathematical Methods for Physicists, 3rd ed.* Orlando, FL: Academic Press,
pp. 530-534, 1985.

© 1996-9

1999-05-26