Definition:Linear Combination of Subsets of Vector Space/Finite Case

Definition

Let $K$ be a field.

Let $X$ be a vector space over $K$.

Let $n \in \N$.

Let $E_1, E_2, \ldots, E_n$ be subsets of $X$ and $\lambda_1, \lambda_2, \ldots, \lambda_n \in K$.

We define the linear combination $\ds \sum_{i \mathop = 1}^n \lambda_i E_i$ by:

$\ds \sum_{i \mathop = 1}^n \lambda_i E_i = \set {\sum_{i \mathop = 1}^n \lambda_i x_i : x_i \in E_i \text { for each } i \in \set {1, 2, \ldots, n} }$