Linear Span

Definition 1. 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 $S$.

Proposition 1. 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

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