Exercise 17, § 9

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