Generalisation of the following

Question

Here, the codomain is itself the field of scalars. So, it was easy to write in that form. In Friedberg, we are also asked for an analogous result when the codomain is $\mathbb{R}^m$ instead of $\mathbb{R}$. I tried to get a composition of two transformations $T:\mathbb{R}^n\rightarrow\mathbb{R}^m$ and $S:\mathbb{R}^m\rightarrow\mathbb{R}$. So, $S$o$T$ and $S$ is known in the above form. Is this the right approach? What exactly is the analogous statment?