Subspace

Let $\Bbb{V}$ be a Real Vector Space (e.g., the real continuous functions $C(I)$ on a Closed Interval $I$, 2-D Euclidean Space $\Bbb{R}^2$, the twice differentiable real functions $C^{(2)}(I)$ on $I$, etc.). Then $\Bbb{W}$ is a real Subspace of $\Bbb{V}$ if $\Bbb{W}$ is a Subset of $\Bbb{V}$ and, for every ${\bf w}_1$, ${\bf w}_2 \in \Bbb{W}$ and $t\in\Bbb{R}$ (the Reals), ${\bf w}_1+{\bf w}_2 \in \Bbb{W}$ and $t{\bf w}_1 \in \Bbb{W}$. Let $(H)$ be a homogeneous system of linear equations in $x_1$, ..., $x_n$. Then the Subset $S$ of $\Bbb{R}^n$ which consists of all solutions of the system $(H)$ is a subspace of $\Bbb{R}^n$.

More generally, let $F_q$ be a Field with $q=p^\alpha$, where $p$ is Prime, and let $F_{q,n}$ denote the $n$-D Vector Space over $F_q$. The number of $k$-D linear subspaces of $F_{q,n}$ is

$$N(F_{q,n})={n\choose k}_q,$$

where this is the q-Binomial Coefficient (Aigner 1979, Exton 1983). The asymptotic limit is

$$N(F_{q,n})=\cases{ c_e q^{n^2/4}[1+o(1)] & for $n$\ even\cr c_o q^{n^2/4}[1+o(1)] & for $n$\ odd,\cr}$$

where

$$c_e = {\sum_{k=-\infty}^\infty q^{-k^2}\over\prod_{j=1}^\infty (1-q^{-j})}$$

$$c_o = {\sum_{k=-\infty}^\infty q^{-(k+1/2)^2}\over\prod_{j=1}^\infty (1-q^{-j})}$$

(Finch). The case $q=2$ 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. $q$-Hypergeometric Functions and Applications. New York: Halstead Press, 1983.

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