Affine space/Product with K/Standard space/Basis/Exercise

Let ${}V$ be a ${}K$-vector space. We consider the set

${}E={\left\{(v,1)\mid v\in V\right\}}\subset V\times K\,,$

which is an affine space over ${}V$.

a) Show that the points ${}P_{i}=(v_{i},1)$, ${}i=1,\ldots ,n$, form an affine basis of ${}E$ if and only if the ${}P_{i}$ (considered as vectors in ${}V\times K$) form a vector space basis of ${}V\times K$.

b) Show that, in this case, for a point ${}P\in E$, the barycentric coordinates of ${}P$ with respect to ${}P_{1},\ldots ,P_{n}$ equal the coordinates of ${}P$ with respect to the vector space basis ${}P_{1},\ldots ,P_{n}$.

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