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

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

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