Span - Imported Knowlpedia Content

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

\operatorname{span}(A):=\bigcap\{\,Y\subset X \mid Y \text{ is a linear subspace of }X\text{ and }A\subset Y\,\}.

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

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).

Remarks

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