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

$\phi :K⟶K,x⟼\phi (x),$

is determined (by fact, but this is also directly clear) by $\phi (1)$, or by the value $\phi (t)$ for a single element $t\in K$, $t\ne 0$. In particular, $\phi (x)=ax$, with a uniquely determined $a\in K$. In the context of physics, for $K=ℝ$, 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“.