Subspaces

In linear algebra, a subspace is a subset W of a vector space V over a field F that is itself a vector space under the same operations as V. Equivalently, W must contain the zero vector of V and be closed under addition and scalar multiplication.

Examples include the zero subspace {0}, the space V itself, and any line through the origin in R^n. More generally, the span of any subset S of V is a subspace, and the intersection of any collection of subspaces is also a subspace.

Subspaces have a notion of dimension: the dimension of a subspace W is the size of a basis for W. Every subspace has a basis, and in a finite-dimensional V, every subspace W is finite-dimensional with dim W ≤ dim V. The sum of subspaces W1 and W2, denoted W1 + W2, is the smallest subspace containing both, while their intersection W1 ∩ W2 is their common part. An important relation for finite-dimensional spaces is dim(W1 + W2) = dim W1 + dim W2 − dim(W1 ∩ W2).

Subspaces can be characterized in several ways. They are precisely the kernels (null spaces) of linear maps from V to another vector space, or equivalently the solution sets to homogeneous systems of linear equations.

Not every subset is a subspace. A line not passing through the origin, for example, is not closed under scalar multiplication and thus is not a subspace. Similarly, a set that does not contain the zero vector cannot be a subspace.