One-dimensional extension of a rational-linear map
= One-dimensional extension of a rational-linear map
{title2=$h(d+qt)=f(d)+qa$}
Let $D$ be a rational <vector subspace> of the <real numbers> and let $t\notin D$. Every element of $D+\mathbb Qt$ has a unique expression $d+qt$. For any prescribed <real number> $a$, the rule $h(d+qt)=f(d)+qa$ extends a <rational-linear map> $f$ on $D$ and remains rational-linear. The uniqueness follows because a nonzero rational multiple of $t$ cannot lie in $D$.