Definition:Linear Combination of Subsets of Vector Space

Theorem
Let $K$ be a field.

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

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

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


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

When $\mu = 0$, we write $\lambda A + 0 B = \lambda A$ and call this the dilation of $A$ by $\lambda$.