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Subspace
Let 
be a real vector
space (e.g., the real continuous functions 
on a closed
interval 
, two-dimensional 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 
-dimensional vector space over 
. The number
of 
-D linear subspaces of 
is
 |
(1)
|
where this is the q-binomial coefficient (Aigner 1979, Exton 1983). The asymptotic limit is
![N(F_(q,n))={c_eq^(n^2/4)[1+o(1)] for n even; c_oq^(n^2/4)[1+o(1)] for n odd,](/web/20140325055540im_/http://mathworld.wolfram.com/images/equations/Subspace/NumberedEquation2.gif) |
(2)
|
where
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
(Finch 2003), where 
is a Jacobi
theta function and 
is a q-Pochhammer
symbol. The case 
gives the q-analog
of the Wallis formula.
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