# Definition:Generated Submodule/Linear Span

## Definition

Let $K$ be a division ring or a field.

Let $V$ be a vector space over $K$.

Let $A \subseteq V$ be a subset of $V$.

Then the **linear span of $A$**, denoted $\span A$ or $\map \span A$, is the set of all linear combinations (of finite length) of vectors in $A$.

The **linear span of $A$** is formally defined as:

- $\map \span A = \ds \set {\sum_{i \mathop = 1}^n \alpha_i v_i: n \in \N_{\ge 1}, \alpha_i \in K, v_i \in A}$

## Linear Span in Real Vector Space

Let $n \in \N_{>0}$.

Let $\R^n$ be a real vector space.

Let $A \subseteq \R^n$ be a subset of $\R^n$.

Then the **linear span of $A$**, denoted $\span A$ or $\map \span A$, is the set of all linear combinations (of finite length) of vectors in $A$.

In the case where $A$ is a finite subset of $\R_n$ such that:

- $A = \set {\mathbf v_1, \mathbf v_2, \dotsc, \mathbf v_k}$

for some $k \in \N_{>0}$, the **linear span of $A$** is formally defined as:

- $\ds \map \span {\mathbf v_1, \mathbf v_2, \dotsc, \mathbf v_k} = \set {\sum_{i \mathop = 1}^k \alpha_i \mathbf v_i: 1 \le i \le k, \alpha_i \in \R, \mathbf v_i \in A }$

## Also denoted as

One also frequently encounters the notation $\sequence A$.

Typically, when $A$ is small, this is also written by substituting the braces for set notation by angle brackets.

For example, when $A = \set {x_1, x_2}$, one writes $\sequence {x_1, x_2}$ for $\span A$.

On this site, the notations using $\span$ are preferred, so as to avoid possible confusion.

## Also see

## Sources

- 1994: Robert Messer:
*Linear Algebra: Gateway to Mathematics*: $\S 4.4$ - For a video presentation of the contents of this page, visit the Khan Academy.