$\mathbb{Q}$ - linear function from $\mathbb{R}$ to $\mathbb{R}$ with kernel $\mathbb{Q}$ I read somewhere that it is possible to construct a $\mathbb{Q} $ - linear function $f:\mathbb{R} \rightarrow \mathbb{R}$ with $\ker f = \mathbb {Q}$ . Can someone enlighten me on that matter? I thought it might be an easy matter of extending the canonic quotient map $\mathbb{R} \rightarrow \mathbb{R}/\mathbb{Q}$ with an isomorphism  $\mathbb{R}/\mathbb{Q}  \rightarrow \mathbb{R}$, but that is probably nothing more than going round in circles.

More of the same:
Assume you have $f$ a non-zero $\mathbb{Q}$-linear map from $\mathbb{R}$ to $\mathbb{Q}$. Consider a $\beta$ so that $\alpha\colon=f(\beta)\ne 0$. The map     $$p\colon \mathbb{R} \to \mathbb{Q}\\
x\mapsto \frac{1}{\alpha} f(\beta x)$$  is a $\mathbb{Q}$-linear projection. Then $p'\colon= 1_{\mathbb{R}}-p$ has kernel $\mathbb{Q}$. 
