# Linear mapping/Linear combinations/Exercise

Let ${\displaystyle {}K}$ be a field and let ${\displaystyle {}V}$ and ${\displaystyle {}W}$ be ${\displaystyle {}K}$-vector spaces. Let
${\displaystyle \varphi \colon V\longrightarrow W}$
be a linear map Prove that for all vectors ${\displaystyle {}v_{1},\ldots ,v_{n}\in V}$ and coefficients ${\displaystyle {}s_{1},\ldots ,s_{n}\in K}$ the relationship
${\displaystyle {}\varphi {\left(\sum _{i=1}^{n}s_{i}v_{i}\right)}=\sum _{i=1}^{n}\lambda _{i}\varphi (v_{i})\,}$