Definition:Linear Combination of Subsets of Vector Space

Definition

Let $K$ be a field.

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

Dilation

Let $E$ be a subset of $X$ and let $\lambda \in K$.

We define the dilation of $E$ by $\lambda$, written $\lambda E$, by:

$\lambda E = \set {\lambda x : x \in E}$

Binary Case

Let $A$ and $B$ be subsets of $X$ and $\lambda, \mu \in K$.

We define the linear combination $\lambda A + \mu B$ by:

$\lambda A + \mu B = \set {\lambda a + \mu b : a \in A, \, b \in B}$

Finite Case

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:

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