Subspace
Let 
be a Real Vector Space (e.g., the real continuous functions 
on aClosed 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 theSubset 
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 
-DVector Space over . The number of 
-D linear subspaces of is
where this is the q-Binomial Coefficient (Aigner 1979, Exton 1983). The asymptotic limit is
where
 |  |  | |
 | |  |
(Finch). The case
gives the q-Analog of the Wallis Formula.See also q-Binomial Coefficient, Subfield, Submanifold
References
Aigner, M. Combinatorial Theory. New York: Springer-Verlag, 1979. Exton, H. Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/dig/dig.html
-Hypergeometric Functions and Applications. New York: Halstead Press, 1983.
- Barth Sextic
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