# Linear mapping/Field/Definition

Linear mapping

Let be a field, and let and be -vector spaces. A mapping

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

- for all .
- for all and .

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.

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