Suppose $G$ is a reductive group with maximal torus $T$, and $U$ is a $T$-stable subgroup. For $\alpha$ a root relative to $T$, fix an isomorphism $u_\alpha\colon\mathbb{G}_a\to U_\alpha$ from the additive group to the root subgroup. I read that if $u_\alpha(c)\in U$ for a $c\neq 0$, then $U_\alpha\subseteq U$ since $tu_\alpha(c)t^{-1}=u_\alpha(\alpha(t)c)\in U$ and $\alpha\neq 0$.
Since $U_\alpha=\operatorname{im}(u_\alpha)$, how do we know that $\alpha(t)c$, or equivalently just $\alpha(t)$, ranges over all of $\mathbb{G}_a$ as $t$ ranges over $T$? Are nontrivial characters of a torus always surjective?