Dilation of Subset of Vector Space Distributes over Sum

Theorem

Let $K$ be a field.

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

Finite Case

Let $A_1, \ldots, A_n \subseteq X$ and $\lambda \in \GF$.

Then:

$\ds \lambda \sum_{j \mathop = 1}^n A_j = \sum_{j = 1}^n \paren {\lambda A_j}$

where:

$\ds \lambda \paren \ldots$ denotes dilation by $\lambda$

$\ds \sum_{j \mathop = 1}^n A_j$ denotes the linear combination of subsets of a vector space.

General Case

Let $\family {E_\alpha}_{\alpha \mathop \in A}$ be an $A$-indexed family of sets.

Let $\lambda \in K$.

Then:

$\ds \lambda \sum_{\alpha \mathop \in A} E_\alpha = \sum_{\alpha \mathop \in A} \paren {\lambda E_\alpha}$