Let's suppose that $S$ and $T$ are sets, $S$ is finite, cardinality of $T$ is larger than that of $S$ (include the possibility where T could be an infinite set), Then, define $f: \mathbb{R}\langle T\rangle \rightarrow \mathbb{R}\langle S\rangle $ is a linear map. Show that $kerf \neq 0$.

My approach to the problem reduces to $T$ being finite. Then, when I tried the case of $T$ being infinite I tried to define a bijection to polynomials (where I proved something similar) , but I didn't succeed. Do you have any ideas in how to proceed. Thanks