Linear mapping/Field/Definition

Linear mapping

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

$\phi :V⟶W$

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

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