Span

Definition

The set of all linear_combinations of a list of vectors $v_{1},\ldots,v_{m}$ in $V$ is called the span of $v_{1},\ldots,v_{m}$ denoted $\text{span }\left. \left( v_{1},\ldots,v_{m} \right) \right.$. In other words, pick

$$ \text{ span }\left. \left( v_{1},\ldots,v_{m} \right) \right. = \{ a_{1}v_{1} + \cdots + a_{m}v_{m}:a_{1},\ldots,a_{m} \in \mathbb{F}\} $$

The span of the empty list $\left. () \right.$ is defined to be $\{ 0\}$.

If $\text{span }\left. \left( v_{1},\ldots,v_{m} \right) \right.$ equals $V$, we say that $v_{1},\ldots,v_{m}$ spans $V$.