Definition:Generated Submodule/Linear Span

Definition
Let $V$ be a vector space over $K$, and let $A \subseteq V$ be a subset.

Then the linear span of $A$, denoted $\operatorname{span} A$ or $\operatorname{span} \left({A}\right)$, is the set


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

It is a linear subspace of $V$, as proved in Linear Span is Linear Subspace.

When dealing with a finite set $A$ of vectors, the linear span can be interpreted as the set of all linear combinations of these vectors.

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


 * $\displaystyle \operatorname{span}\left({\mathbf{v_1},\mathbf{v_2},\cdots,\mathbf{v_k}}\right) = \left\{ {\sum_{i=1}^k \ c_i \ \mathbf{v_i}: c_i \in \R, \mathbf{v_i}\in \R^n, 1 \le i \le k} \right \}$

Alternative Notation
One also frequently encounters $\left\langle{A}\right\rangle$.

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

For example, when $A = \left\{{x_1, x_2}\right\}$, one writes $\left\langle{x_1, x_2}\right\rangle$ for $\operatorname{span} A$.

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