Affine space/Product with K/Realization/Exercise

Let ${}E$ be a nonempty affine space over a ${}K$-vector space ${}V$. Let ${}P\in E$ be a fixed point, and let

$\theta \colon V\longrightarrow E,v\longmapsto P+v,$

be the corresponding bijection. Using this bijection, we identify ${}E$ with

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

via the mapping

$\varphi \colon E\longrightarrow E',P\longmapsto (\theta ^{-1}(P),1).$

a) Show that ${}E'$ is an affine subspace of ${}V\times K$, with translation space ${}V\times 0$.

b) Show that

${}\varphi (Q+v)=\varphi (Q)+v\,$

holds for all ${}Q\in E$.

Create a solution