Subspaces

What is a Subspace?

A subspace is a subset $W$ of a vector space $V$ that is itself a vector space under the same operations. Most action in linear algebra—solution sets, spans, kernels—happens inside subspaces.

Checking every axiom is tedious. Instead, use the Subspace Test: a nonempty subset $W$ is a subspace if it is closed under two operations:

Setting $c = 0$ reveals a vital shortcut: every subspace must contain the zero vector ($\mathbf{0}$). Any subset missing $\mathbf{0}$ fails instantly.

Geometry of Subspaces

In $\mathbb{R}^3$, the only possible subspaces are strict geometric categories:

Common pitfall: A line or plane that is offset from the origin is not a subspace. It fails because it lacks $\mathbf{0}$, and scaling any of its points by $0$ lands at the origin, off the line.

Subspaces also include homogeneous solution sets $A\mathbf{x} = \mathbf{0}$ (the null space), spans, and linear map ranges.