I am reading the book by Ernest Vinberg, 'linear representations of groups'. On page 78, he considers the representation of SU(2). The function space $V_n$ he takes is the space of homogeneous polynomials of degree n in the variables $u_1$ and $u_2$. Then he defines a representation $\Phi$ of the group $SL_2(\mathbb{C})$ in $V_n$ by the rule
$$ (\Phi_n (A)f)(u) = f(u A), $$
where $u = (u_1,u_2)$, $A\in SL_2(\mathbb{C})$, and $f\in V_n $. In more detail, if $A=[a_{ij}]$, then
$$ (\Phi_n (A)f)(u_1, u_2) = f(a_{11}u_1+ a_{21}u_2, a_{12}u_1+ a_{22}u_2) .$$
He then states that
"This definition agrees with the general rule by which transformations act on functions if one regards $u_1$ and $u_2$ as coordinates on the space $U$ dual to the space of column vectors in which $SL_2(\mathbb{C})$ acts naturally, and one accordingly regards $V_n$ as a space of functions on $U$."
I checked that this definition indeed aligns with the association law. But it still looks strange to me. Generally, if we have a linear transform $A$ on a vector space, the induced transform of a function $f$ is defined as
$$ (\Psi(A)f )(r) = f(A^{-1} r) .$$
This definition is very intuitive. $A$ moves the point $A^{-1}r$ to $r$, and in doing so it carries the value of $f$ at $A^{-1}r$ to the point $r$.
In contrast, Vinberg's definition is not so intuitive. But why does it work? What is the relation between his definition and the more intuitive definition?