Let V be a real vector space (e.g., the real continuous functions C(I) on a closed interval I, two-dimensional Euclidean space R^2, the twice differentiable real functions C^((2))(I) on I, etc.). Then W is a real subspace of V if W is a subset of V and, for every w_1, w_2 in W and t in R (the reals), w_1+w_2 in W and tw_1 in W. Let (H) be a homogeneous system of linear equations in x_1, ..., x_n. Then the subset S of R^n which consists of all solutions of the system (H) is a subspace of 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-dimensional vector space over F_q. The number of k-D linear subspaces of F_(q,n) is

 N(F_(q,n))=(n; k)_q,

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,



(Finch 2003), where theta_n(q) is a Jacobi theta function and (q)_infty=(q;q)_infty is a q-Pochhammer symbol. The case q=2 gives the q-analog of the Wallis formula.

See also

q-Binomial Coefficient, Subfield, Submanifold Explore this topic in the MathWorld classroom

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Aigner, M. Combinatorial Theory. New York: Springer-Verlag, 1979.Exton, H. q-Hypergeometric Functions and Applications. New York: Halstead Press, 1983.Finch, S. R. "Lengyel's Constant." Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 316-321, 2003.

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Cite this as:

Weisstein, Eric W. "Subspace." From MathWorld--A Wolfram Web Resource.

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