Let be a Real Vector Space (e.g., the real continuous functions on a Closed Interval , 2-D Euclidean Space , the twice differentiable real functions on , etc.). Then is a real Subspace of if is a Subset of and, for every , and (the Reals), and . Let be a homogeneous system of linear equations in , ..., . Then the Subset of which consists of all solutions of the system is a subspace of .

More generally, let be a Field with , where is Prime, and let denote the -D
Vector Space over . The number of -D linear subspaces of is

where this is the

where

(Finch). The case gives the

**References**

Aigner, M. *Combinatorial Theory.* New York: Springer-Verlag, 1979.

Exton, H. *-Hypergeometric Functions and Applications.* New York: Halstead Press, 1983.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/dig/dig.html

© 1996-9

1999-05-26