span

The span of a set of vectors ${\overrightarrow{𝐯}}_1,\mathrm{\dots },\overrightarrow{{𝐯}_{𝐧}}$ of a vector space $V$ over a field $K$ is the set of linear combinations $a_1{\overrightarrow{𝐯}}_1+\mathrm{\dots }+a_n{\overrightarrow{𝐯}}_n$ with $a_i\in K$. It is denoted $\mathrm{Sp}({\overrightarrow{𝐯}}_1,\mathrm{\dots },{\overrightarrow{𝐯}}_n)$. More generally, the span of a set $S$ (not necessarily finite) of vectors is the collection of all (finite) linear combinations of elements of $S$. The span of the empty set is defined to be the singleton consisting of the zero vector $\overrightarrow{\mathrm{𝟎}}$.

For example, the standard basis vectors $\widehat{ı}$ and $\widehat{ȷ}$ span ${ℝ}^2$ because every vector of ${ℝ}^2$ can be represented as a linear combination of $\widehat{ı}$ and $\widehat{ȷ}$.

$\mathrm{Sp}({\overrightarrow{𝐯}}_1,\mathrm{\dots },{\overrightarrow{𝐯}}_n)$ is a subspace of $V$ and is the smallest subspace containing ${\overrightarrow{𝐯}}_1,\mathrm{\dots },{\overrightarrow{𝐯}}_n$.

Span is both a noun and a verb; a set of vectors can span a vector space, and a vector can be in the span of a set of vectors.

Checking span: To see whether a vector is in the span of other vectors, one can set up an augmented matrix, since if $\overrightarrow{𝐮}$ is in the span of ${\overrightarrow{𝐯}}_1,{\overrightarrow{𝐯}}_2$, then $\overrightarrow{𝐮}=x_1{\overrightarrow{𝐯}}_1+x_2{\overrightarrow{𝐯}}_2$. This is a system of linear equations. Thus, if it has a solution, $\overrightarrow{𝐮}$ is in the span of ${\overrightarrow{𝐯}}_1,{\overrightarrow{𝐯}}_2$. Note that the solution does not have to be unique for $\overrightarrow{𝐮}$ to be in the span.

To see whether a set of vectors spans a vector space, you need to check that there are at least as many linearly independent vectors as the dimension of the space. For example, it can be shown that in ${ℝ}^n$, $n+1$ vectors are never linearly independent, and $n-1$ vectors never span.

Remark. We can define the concept of span also for a module $M$ over a ring $R$.Given a subset $X\subset M$ we define the module generated by $X$ as the set of all finite linear combinations of elements of $X$.Be aware that in general there does not exist a linearly independent subset which generates the whole module, i.e. there does not have to exist a basis.Also, even if $M$ is generated by $n$ elements, it is in general not true that any other set of $n$ linearly independent elements of $M$ spans $M$.For example $\mathbb{Z}$ is generated by $1$ as a $\mathbb{Z}$-module but not by $2$.