Exercise 17, § 9

Let $V$ be a vector space of finite dimension $n$ over a field $K$, $(e_i)_{1 \leq i \leq n}$ a basis of $V$, $K'$ a subfield of $K$ and $V'$ the $n$-dimensional vector space over $K'$ generated by the $e_i$. Give an example of an injective mapping of $V'$ into $V$, mapping any three points on a straight line of the affine space $V'$ to three points on a straight line in the affine space $V$, but not necessarily mapping two parallel lines to sets contained in two parallel lines. (Embed $V$ in the projective space $E$ canonically associated with it and consider a projective transformation $u$ of $E$ into itself such that the inverse image under $u$ of the hyperplane at infinity is distinct from this hyperplane and contains no point of $V'$; for example $K$ can be taken to be infinite and $K'$ finite.)