Real Subspaces

Definition

A nonempty set $S \subset \R ^n$ is called a subspace (or linear subspace, vector subspace) if

1. $x + y \in S$ for all $x, y \in S$, and
2. $\alpha x \in S$ for all $\alpha \in \R $, $x \in S$.

Examples

The set $S_1 = \R ^n$ is a subspace. In other words, the entire set is a subspace of itself. The set $S_2 = \set{0}$, consisting of a single point, the origin, is a subspace. $S_1$ is the biggest subspace. In other words, if $S'$ is another subspace of $\R ^n$, then $S' \subset S_1$. If $S$ is a subspace, it is nonempty, so there is $x \in S$, and it is closed under scalar multiplication, so $0\cdot x = 0 \in S$. In other words, every subspace contains the origin. So $S_2$ is the smallest subspace, in the sense that if $S'$ is another subspace $S_2 \subset S'$.

The span (see Real Vectors Span) of a set of vectors $v_1, \dots , v_k$ is a subspace. For two subspaces $S, T \subset \R ^n$, their sum

\[ S + T = \Set{x + y}{x \in S, y \in T} \]

Geometric intuition

Roughly speaking, a subspace $S$ is a flat set which passes through the origin. In $\R ^2$, the subspaces are the lines. In $\R ^3$, the lines and the planes.