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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

 (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.

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

## References

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.

Subspace

## Cite this as:

Weisstein, Eric W. "Subspace." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Subspace.html