Linear mapping/One-dimensional/Proportionality and rule of three/Example

The easiest linear mappings are (beside the null mapping) the linear maps from ${}K$ to ${}K$. Such a linear mapping

$\varphi \colon K\longrightarrow K,x\longmapsto \varphi (x),$

is determined (by fact, but this is also directly clear) by ${}\varphi (1)$, or by the value ${}\varphi (t)$ for an single element ${}t\in K$, ${}t\neq 0$. In particular, ${}\varphi (x)=ax$, with a uniquely determined ${}a\in K$. In the context of physics, for ${}K=\mathbb {R}$, and if there is a linear relation between two measurable quantities, we talk about proportionality, and ${}a$ is called the proportionality factor. In school, such a linear relation occurs as "rule of three“.