Suppose that I have a very structured vector of the form:
$$v(x)=\begin{bmatrix}1 & x & x^2 & \cdots & x^n\end{bmatrix}^\top$$
where $x \in \mathbb{R}$. More in general $v$ might be a vector where each entry is a function of $x$, i.e., $v(x)= [f_1(x) \; f_2(x) \; \ldots f_n(x)]^\top$.
Suppose now that there exists a specific linear transformation $T$ which maps the very structured vector $v(x)$ to another very structured vector $w(y)$, i.e., :
$$Tv(x)= \begin{bmatrix}1 & y & y^2 & \cdots & y^n\end{bmatrix}^\top $$
Is there a name for such "structure-preserving" linear transformations? Also, I have found that for a specific matrix $T$ (which is highly structured), I can obtain the relation above. How can I prove that I cannot find a more general matrix for which the relationship above hold? What is the procedural way to show it? For contradiction?