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
\[\langle S \rangle =% \left\{ \sum_{i = 1}^n \lambda_i v_i : \lambda_1, \cdots, \lambda_n \in F \right\}.\]