Definition:Generated Submodule/Linear Span

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

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

Then the linear span of $A$, denoted $\operatorname{span} A$ or $\map {\operatorname{span} } A$, is the set:


 * $\displaystyle \set {\sum_{k \mathop = 1}^n \alpha_k f_k: n \in \N_{\ge 1}, \alpha_i \in K, f_i \in A}$

The linear span can be interpreted as the set of all linear combinations (of finite length) of these vectors.

Definition for $\R^n$
In $\R^n$ (where $n \in \N_{>0}$), above definition translates to:


 * $\displaystyle \map {\operatorname{span} } {\mathbf v_1, \mathbf v_2, \dotsc, \mathbf v_k} = \set {\sum_{i \mathop = 1}^k \ c_i \ \mathbf v_i: c_i \in \R, \mathbf v_i \in \R^n, 1 \le i \le k}$

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 $\operatorname{span} A$.

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

Also see

 * Definition:Spanning Set
 * Definition:Closed Linear Span
 * Linear Span is Linear Subspace

Generalizations

 * Definition:Generated Submodule