Let $C$ an elliptic curve over $\mathbb Q$. Assume that the rank of $C(ℚ)$ is equal to $r$. Then the cardinality of a maximal independent set in $C(ℚ)$ is $r$, thus there exists $r$ independent points ${P_1,P_2,\dots,P_r}$ of infinite order in $C(ℚ)$, i.e., $P_{k}=(x_{k},y_{k})\in\mathbb Q^2,\ k=1,..,r$ (we ignor the torsion part of $C(ℚ)$ to use the canonical height) such that if $∑_{k=1}^{r}α_{k}P_{k}=0$, then $α_{k}=0$ for all $k=1,..,r$.
My question is: Is it possible to say that every point $P$ in $C(ℚ)$ other than ${P_1,P_2,\dots,P_r}$ is of finite order?