Vector spaces can contain other vector spaces. If $V$ is a vector space, then $S \subseteq V$ is said to be a subspace of $V$ if

(i) $\mathbf{0} \in S$

(ii) $S$ is closed under addition: $\mathbf{x}, \mathbf{y} \in S$ implies $\mathbf{x}+\mathbf{y} \in S$

(iii) $S$ is closed under scalar multiplication: $\mathbf{x} \in S, \alpha \in \mathbb{R}$ implies $\alpha \mathbf{x} \in S$

Note that $V$ is always a subspace of $V$, as is the trivial vector space which contains only 0 .

As a concrete example, a line passing through the origin is a subspace of Euclidean space.

Some of the most important subspaces are those induced by linear maps. If $T: V \rightarrow W$ is a linear map, we define the nullspace $^{2}$ of $T$ as

$$

\operatorname{null}(T)=\{\mathbf{x} \in V \mid T \mathbf{x}=\mathbf{0}\}

$$

and the range (or the columnspace if we are considering the matrix form) of $T$ as

$$

\operatorname{range}(T)=\{\mathbf{y} \in W \mid \exists \mathbf{x} \in V \text { such that } T \mathbf{x}=\mathbf{y}\}

$$

It is a good exercise to verify that the nullspace and range of a linear map are always subspaces of its domain and codomain, respectively.