Let $T : \mathbb{R}[x]_{\leq 1} \rightarrow \mathbb{R}[x]_{\leq 2}$ be the linear map defined by the differential operator $T = (x-2) \frac{d}{d x} - \text{id}$.
Now my exercise asks me to find the matrix representing $T$ with respect to the standard coordinate systems
$(1 \; x): \mathbb{R}^2 \rightarrow \mathbb{R}[x]_{\leq 1} \quad \text{ and } \quad (1 \; x \; x^2) : \mathbb{R}^3 \rightarrow \mathbb{R}[x]_{\leq 2}$
I guess what I am mostly confused about, is the order in which I should compose the maps. Am I looking for a map from $\mathbb{R}^2$ to $\mathbb{R}^3$, or am I looking for a map from $\mathbb{R}[x]_{\leq 1}$ to $\mathbb{R}[x]_{\leq 2}$?
Currently, the more I think about it, the more confused I get, so I am looking for a little clarity most of all.