Vector subspace
In linear algebra and related fields of mathematics, a linear subspace, also known as a vector subspace, or, in the older literature, a linear manifold, is a vector space that is a subset of some other (higher-dimension) vector space. A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of .
Let K be a field (such as the real numbers), V be a vector space over K, and let W be a subset of V. Then W is a subspace if:
Example I: Let the field K be the set R of real numbers, and let the vector space V be the real coordinate space R3. Take W to be the set of all vectors in V whose last component is 0. Then W is a subspace of V.
Proof:
Example II: Let the field be R again, but now let the vector space be the Cartesian plane R2. Take W to be the set of points (x, y) of R2 such that x = y. Then W is a subspace of R2.
Proof:
In general, any subset of the real coordinate space Rn that is defined by a system of homogeneous linear equations will yield a subspace. (The equation in example I was z = 0, and the equation in example II was x = y.) Geometrically, these subspaces are points, lines, planes, and so on, that pass through the point 0.
Example III: Again take the field to be R, but now let the vector space V be the set RR of all functions from R to R. Let C(R) be the subset consisting of continuous functions. Then C(R) is a subspace of RR.
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