Suppose $(u,v) = A(x,y)$ is affine transformation.
Where $u = ax + by + e$, and $v = cx + dy + f$ , and the inverse transformation given by $x = a'u + b'v + e'$ and $y = c'u + d'v + f'$.
Suppose further that $C$ is a curve given by the polynomial equation $P(x,y) = 0$.
I am wondering how to interpret the image of $C$ under the affine map $A$.
It seems that $(q,r)$ is a point of the curve $C$ (and so $P(q,r) = 0$) if and only if $A(q,r) = (aq + br + e, cq + dr + f)$ is a point of the curve defined by the "transformed polynomial" $P(u,v) = P(a'u + b'v + e',c'u + d'v + f')$.
Is this correct?