Span

Let $X$ be a vector space and let $A\subset X$.

The linear subspace generated by $A$, also called the span of $A$, is defined as

Equivalently, it is the intersection of all subspaces containing $A$, hence the smallest subspace that contains $A$.

A central theorem (see span as linear combinations ) identifies $\operatorname{span}(A)$ with all finite linear combinations of elements of $A$.

Examples:

• In $\mathbb{R}^2$, $\operatorname{span}\{(1,0)\}$ is the $x$-axis.
• In $\mathbb{R}^3$, $\operatorname{span}\{(1,0,0),(0,1,0)\}$ is the $xy$-plane.
• If $A=\emptyset$, then $\operatorname{span}(A)=\{0\}$ (intersection of all subspaces).