Linear mapping/Field/Definition/Explanations/Remark

Here, the first property is called additivity and the second property is called compatibility with scaling. When we want to stress the base field, then we say ${}K$-linearity. The identity ${}\operatorname {Id} _{V}\colon V\rightarrow V$, the null mapping ${}V\rightarrow 0$ and the inclusion ${}U\subseteq V$ of a linear subspace are the simplest examples for linear mappings. For a linear mapping, the compatibility with arbitrary linear combination holds, that is,

${}\varphi {\left(\sum _{i=1}^{n}s_{i}v_{i}\right)}=\sum _{i=1}^{n}s_{i}\varphi (v_{i})\,,$

see exercise.