Let $R$ be a D273: Division ring such that

(i)	$V$ and $W$ are each a D29: Vector space over $R$
(ii)	$f : V \to W$ is a D18: Map from $V$ to $W$

Then $f$ is a linear map from $V$ to $W$ over $R$ if and only if \begin{equation} \forall \, N \in \{ 1, 2, 3, \ldots \} : \forall \, x \in V^N : \forall \, r \in R^N : f \left( \sum_{n = 1}^N r_n x_n \right) = \sum_{n = 1}^N r_n f(x_n) \end{equation}