Scaled Sets

Definition

The scalar multiple of a subset of a vector space by a given scalar is the set of all vectors which are the scalar product of the given scalar and the vector.

The symmetric reflection of a subset of a vector space is the scalar multiple by the additive inverse of the field. A subset of a vector space is symmetric if it is its own symmetric reflection.

Notation

Let $(V, \F )$ be a vector space. Let $M \subset V$ and $\lambda \in \F $. The scalar multiple of $M$ by $\lambda $ is the set

\[ \Set*{\lambda x}{x \in M}, \]

which we denote by $\lambda M$.

The symmetric refletion of $M$ is $(-1)M$ which we denote by $-M$. $M$ is symmetric if $M = -M$.