Affine space/Finite point family/Mapping to affine hyperplane/Exercise

Let ${}E$ be an affine space, and let ${}P_{1},\ldots ,P_{n}$ be a finite family of points in ${}E$. Let

${}F={\left\{\left(a_{1},\,\ldots ,\,a_{n}\right)\in K^{n}\mid \sum _{i=1}^{n}a_{i}=1\right\}}\subset K^{n}\,.$

Show that the assignment

$\left(a_{1},\,\ldots ,\,a_{n}\right)\longmapsto \sum _{i=1}^{n}a_{i}P_{i}$

defines a well-defined affine-linear mapping from ${}F$ to ${}E$.

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