Let $V$ be a vector space and $S\subseteq V$ be a subset.
The linear span of $S$ is the generated subspace of $S$:
\[\langle S\rangle = \bigcap\{W\leq V : S\subseteq W\}.\]Each element of $\langle S\rangle$ is called a linear combination of $S$.
Let $V$ be a vector space over $F$. For any finite subset
\[S = \{v_1,\cdots,v_n\} \subseteq V\]($n\geq 1$), we have
\[\langle S\rangle = \left\{\sum_{i=1}^n \lambda_i v_i :% \lambda_1,\cdots,\lambda_n \in F\right\}.\]
Let $V$ be a vector space. For any subset $S\subseteq V$, $\langle S\rangle$ is a subspace.