A subspace is a subset of a vector space that still contains enough structure to be a vector space on its own, using the same addition and scalar multiplication.

For a subset $U\subseteq V$, this means that linear combinations of vectors in $U$ must remain in $U$.

The subspace test

A nonempty subset $U$ is a subspace when, for all $\mathbf u,\mathbf v\in U$ and all scalars $a,b$,

$$a\mathbf u+b\mathbf v\in U.$$

This single closure condition includes closure under addition and scalar multiplication. It also forces the zero vector to belong to $U$.

A geometric example

In $\mathbb R^3$, the set

$$U={(x,y,0):x,y\in\mathbb R}$$

is the $xy$-plane through the origin. Adding two vectors in this plane or scaling one cannot create a nonzero third component, so $U$ is a subspace.

A plane not passing through the origin is not a subspace, because it does not contain the zero vector.

Spans are subspaces

For any collection of vectors,

$$\operatorname{span}(\mathbf v_1,\ldots,\mathbf v_k)$$

is automatically a subspace. Its elements are already defined as all possible linear combinations of the generators.

Once a subspace is identified, it can have its own basis and dimension just like the surrounding vector space.