Subspaces

A rigorous guide to the subspace test, bases, sums, intersections, and affine distinctions.

Vector-space ideas connect sets, operations, spanning, independence, basis, and dimension. A vector space sits at the centre of a dependency map. Subspace conditions lead to span and independence, which meet at basis and determine dimension; kernel and range connect through a transformation.

Precise definition

A subset $W$ of a vector space $V$ is a subspace if it contains zero and is closed under vector addition and scalar multiplication. Equivalently, for all $u,v\in W$ and scalars $a,b$, the combination $au+bv$ lies in $W$.

Notation and mathematical language

Kernels, ranges, spans, solution sets of homogeneous linear systems, and intersections of subspaces are subspaces. A solution set of $Ax=b$ with $b\ne0$ is generally affine rather than a subspace. The sum is $U+W=\{u+w\}$.

Conceptual picture

A subspace is a vector space using inherited operations. The combined linear-combination test is efficient because it verifies zero and both closure properties at once when the set is nonempty.

Fully worked example

Interpretation and application

Subspaces describe feasible directions, conserved constraints, data models, and invariant state sets. Affine feasible sets often arise in applications; translating by one known point converts them to a related homogeneous subspace.