Let $K$ be a field with $\operatorname{char}(K)=0$ and $G$ a connected split reductive group over $K$ with a maximal torus $T$. For $\mathfrak{t}=\operatorname{Lie}(T)$ being the associated Lie algebra, we get by associating to a character $\chi$ of $T$ its differential at the identity, a natural map \begin{equation*} \phi:X_*(T)=\operatorname{Hom}(T,\mathbb{G}_m) \rightarrow \operatorname{Hom}(\mathfrak{t}_K,K)=\mathfrak{t}^{*}. \end{equation*}
For $T$ being split, i.e $T\cong \mathbb{G}_m^n$ for some $n$, we have that $X_*(T)\cong \mathbb{Z}^n$. Is it then possible to describe the image of a tuple $(\lambda_1,\ldots,\lambda_n) \in \mathbb{Z}^n$ under $\phi$?