subspace

A subspace is a vector space tucked inside a bigger one — a smaller arena that is itself a perfectly good place for vectors to live. The xy-plane sitting inside 3D space is a subspace; a single line through the origin inside the plane is a subspace. The key idea is that a subspace is not just any subset; it has to be self-contained in the same way the whole space is.

There is a quick three-part test for whether a subset is a subspace. First, it must contain the zero vector. Second, it must be closed under addition: add any two of its vectors and you stay inside. Third, it must be closed under scaling: multiply any of its vectors by any number and you stay inside. Pass all three and you have a subspace; fail any one and you do not. A practical consequence is that every subspace must pass through the origin, so a line or plane that is shifted away from the origin is not a subspace.

A tiny example: in R^3, the set of all vectors of the form (x, y, 0) — everything with a zero third coordinate — is a subspace, the familiar flat xy-plane. It contains (0, 0, 0), the sum of two such vectors still has a zero third coordinate, and scaling one keeps that zero. But the set of vectors (x, y, 1) is not a subspace: it misses the origin, and adding two of them gives a third coordinate of 2, landing outside.