Subspaces

Vector Spaces Inside Vector Spaces

A subspace is a subset $W$ of a vector space $V$ that is itself a vector space under the same operations. Subspaces are where most of the action in linear algebra happens — solution sets, spans, kernels, and images are all subspaces.

Checking all the axioms would be tedious, but there is a shortcut. A nonempty subset $W$ is a subspace exactly when it is closed under the two operations:

1. Closed under addition — if $u, v \in W$ then $u + v \in W$.
2. Closed under scalar multiplication — if $v \in W$ and $c$ is a scalar then $cv \in W$.

If both hold, all the other axioms are inherited automatically from $V$. A convenient consequence: taking $c = 0$ shows every subspace must contain the zero vector — so a subset missing $\mathbf{0}$ can be rejected immediately.

Geometrically, the subspaces of $\mathbb{R}^3$ are exactly: the zero subspace $\{\mathbf{0}\}$, every line through the origin, every plane through the origin, and all of $\mathbb{R}^3$. Crucially, they must pass through the origin — a line not through the origin is not a subspace, because it fails closure (scaling a point on it by 0 lands off the line, and it lacks the zero vector).

Subspaces arise everywhere: the solutions of a homogeneous system $A\mathbf{x} = \mathbf{0}$ form a subspace (the null space), as do the span of any set of vectors and the range of a linear map. Recognizing a subspace lets you apply all of vector-space theory to it directly.

Common pitfall: thinking a line or plane that does not pass through the origin is a subspace. Every subspace must contain the zero vector and be closed under scaling — so a line offset from the origin fails (scaling any of its points by 0 gives $\mathbf{0}$, which is not on the line). Only lines and planes through the origin (plus $\{\mathbf{0}\}$ and the whole space) are subspaces.