Linear combination

Definition

Let $F$ be a field and let $V$ be a vector space over $F$. Let $v_1, \dots, v_k \in V$. A linear combination of $v_1, \dots, v_k$ is a vector of the form $$a_1 v_1 + a_2 v_2 + \cdots + a_k v_k,$$ where $a_1, \dots, a_k \in F$ are the coefficients. For an arbitrary family $(v_i)_{i \in I}$, a linear combination is a sum $\sum_{i \in I} a_i v_i$ in which only finitely many coefficients $a_i$ are nonzero.