Span of a Finite Family of Vectors

Let $K$ be a field, let $V$ be a vector space over $K$, let $n$ be a natural number, and let $v\in V^{n}$ be an $n$-tuple in $V$, with components $v_{k}$.

The span of $v$ is the set

$$\operatorname{span}(v)=\Bigl\{u\in V\ :\ u=\sum_{k=1}^{n}c_{k}v_{k}\ \text{ for some }c\in K^{n}\Bigr\},$$

where $c_{k}$ denotes the $k$-th component of the $n$-tuple $c$ in $K$, and the sum is the finite sum in $V$ of the $n$-tuple whose $k$-th component is $c_{k}v_{k}$.