The motivation to this question can be found in
How I can express $(x,y)∈G$ by using the $r$ independent points $P_1,P_2,\ldots,P_r$
We know that there is an isomorphism $$φ:ℤ^{r}×C(ℚ)^{\text{tors}}→ℤ^{r}⊕C(ℚ)^{\text{tors}}=C(ℚ)$$ defined by $$(Q,T)→φ(Q,T)=Q+T$$
where $$Q=∑_{k=1}^{r}α_{k}P_{k}$$ and $T$ is a torsion point.
My question is: Can we extend the map $φ$ to $ℝ^{r}×C(ℚ)^{\text{tors}}→C(ℚ)$ as an isomorphism or not?