Vector space/Surjective mapping with structure/Vector space structure/Exercise

Let ${}K$ be a field, ${}V$ a ${}K$-vector space, and ${}M$ a set with a binary operation

$+\colon M\times M\longrightarrow M,$

and a mapping

$\cdot \colon K\times M\longrightarrow M.$

Let

$\varphi \colon V\longrightarrow M$

be a surjective mapping satisfying

$\varphi (x+y)=\varphi (x)+\varphi (y)\,\,{\text{ and }}\,\,\varphi (sx)=s\varphi (x)$

for all ${}x,y\in V$ and ${}s\in K$. Show that ${}M$ is a ${}K$-vector space.

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