Linear Span

linear-algebra

Definition

Linear Span

Let $V$ be a vector space over a field $F$ and $S\subseteq V$. The linear span (or linear hull) of $S$, denoted $[S]$ or $\mathrm{span}(S)$, is the set of all finite linear combinations of vectors from $S$:

$$[S]=\left\{\sum _{i=1}^k{\alpha }_i{v}_i|k\in \mathbb{N},v_i\in S,{\alpha }_i\in F\right\}.$$

$[S]$ is the smallest subspace of $V$ containing $S$.