Linear mapping/Linear combinations/Exercise

Let ${}K$ be a field, and let ${}V$ and ${}W$ be ${}K$-vector spaces. Let

$\varphi \colon V\longrightarrow W$

be a linear map. Prove that for all vectors ${}v_{1},\ldots ,v_{n}\in V$ and coefficients ${}s_{1},\ldots ,s_{n}\in K$, the relationship

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

holds.

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