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Subspace

Basic definition

A subspace is a subset of a vector space that is also itself a vector space. This term can also be used for a subset of a topological space.

Detailed definition

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 element W and t element R (the reals), w_1 + w_2 element W and t w_1 element 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^α, where p is prime, and let F_(q, n) denote the n-dimensional vector space over F_q.

Related terms

q-binomial coefficient | subfield | submanifold

Educational grade level

college level

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