Example

(1) Let E be an A-module and a an element of E; the mapping $\lambda \mapsto \lambda a$ of the A-module $A_s$ into E is a linear mapping $\theta_a$ such that $\theta_a(1) = a.$ (2) Let I be an open interval of the real line $\mathbf{R}$, E the vector space of differentiable real-valued functions on I and F the vector space of all real-valued functions defined on I. The mapping $x \mapsto x'$ which associates with every differentiable function x its derivative is a linear mapping from E to F.

Note that a homothety $x \mapsto \alpha x$ on an A-module E is not necessarily a linear mapping: in other words, the relation $\alpha(\lambda x) = \lambda(\alpha x)$ does not necessarily hold for all $\lambda \in A$ and $x \in E$. This relation is however true when $\alpha$ belongs to the centre of A; $x \mapsto \alpha x$ is then called a central homothety (cf. no. 13).

If $u : E \to F$ is a linear mapping, then, for every family $(x_i)_{i \in I}$ of elements of E and every family $(\lambda_i)_{i \in I}$ of elements of A such that the support of the family $(\lambda_i x_i)_{i \in I}$ is finite,

as follows immediately from (3) and (4G) by induction on the cardinal of the support of the family $(\lambda_i x_i)$.