Linear mapping/Field/Definition

Linear mapping

Let ${}K$ be a field, and let ${}V$ and ${}W$ be ${}K$-vector spaces. A mapping

$\varphi \colon V\longrightarrow W$

is called a linear mapping if the following two properties are fulfilled.

1. ${}\varphi (u+v)=\varphi (u)+\varphi (v)$ for all ${}u,v\in V$.
2. ${}\varphi (sv)=s\varphi (v)$ for all ${}s\in K$ and ${}v\in V$.