Spanning List¶

Definition¶

A spanning list for a vector space $V$ is a finite list $\mathcal{V}=(\boldsymbol{v}_1,\ldots,\boldsymbol{v}_k)$ of vectors such that

V=\operatorname{span}(\mathcal{V}).

Interpretation¶

A spanning list contains enough vectors to generate the entire space: every vector of $V$ is a linear combination of $\boldsymbol{v}_1,\ldots,\boldsymbol{v}_k$ . It may contain redundant vectors, and then a vector can have several such representations.

Three vectors v1, v2 and v3 = v1 + v2 in the plane, with a vector w that can be written as a combination of them in many ways; the list spans the plane but is redundant.

In the figure, $\boldsymbol{v}_1$ lies on the $x_1$ -axis, $\boldsymbol{v}_2$ on the $x_2$ -axis and $\boldsymbol{v}_3$ on the diagonal between them. The list spans $\mathbb{R}^2$ , and the same vector has several representations: $\boldsymbol{w}=2\boldsymbol{v}_1+1.5\boldsymbol{v}_2=0.5\boldsymbol{v}_1+1.5\boldsymbol{v}_3=\boldsymbol{v}_1+0.5\boldsymbol{v}_2+\boldsymbol{v}_3$ $v_{3}$ . Since $\boldsymbol{v}_3=\boldsymbol{v}_1+\boldsymbol{v}_2$ $v_{2}$ , the list is redundant, and $\boldsymbol{v}_3$ can be omitted without changing the span.

Removing redundant vectors one at a time, as long as some vector is a combination of the others, ends with a linearly independent spanning list, which is a basis. So a spanning list is at least as long as a basis: a spanning list of $\mathbb{R}^n$ has at least $n$ vectors. Adding vectors to a spanning list keeps it spanning, while removing one may not.

To show that a list spans $V$ , take an arbitrary $\boldsymbol{v}\in V$ and show that suitable coefficients exist; checking a few vectors is not enough. A spanning list is finite by definition, so a space that has one is finite-dimensional.

Key Results¶

A basis is a spanning list with no linear redundancy.
A finite-dimensional vector space has a finite spanning list.

Examples¶

$((1,0),(0,1),(1,1))$ spans $\mathbb{R}^2$ but is not a basis.

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 2

Aliases: spanning set, generating set Symbols: element of, real numbers

Last updated: 20 October 2026 03:02