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 $X$.

Let $\lambda \in K$.

The dilation of $E$ by $\lambda$ is defined and written as:

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

where $\lambda x$ is the scalar product of $x$ by $\lambda$.

Binary Case

Let $A$ and $B$ be subsets of $X$.

Let $\lambda, \mu \in K$.

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:

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

General Case

Let $I$ be an indexing set.

For each $\alpha \in I$, let $E_\alpha$ be a subset of $X$ and $\lambda_\alpha \in K$.

We define the linear combination $\ds \sum_{\alpha \mathop \in I} \lambda_\alpha E_\alpha$ by:

$\ds \sum_{\alpha \mathop \in I} \lambda_\alpha E_\alpha = \set {\sum_{i \in F} \lambda_i x_i : F \text { is a finite subset of } I, \, x_i \in E_i \text { for each } i \in I}$