One of the problems in some work I'm doing tells me to consider $GL_2$ acting on $\mathbb{C}[X,Y]$, induced by the natural representation of $GL_2$ on $\mathbb{C}^2$. I just wanted to check 2 things: firstly, is $\mathbb{C}[X,Y]$ typically the set of sums of the form $\sum_{i,j \geq 0}a_{i\,j}X^i Y^j$ $(*)$?
Secondly, what is this action induced by the natural representation on $\mathbb{C}^2$? Obviously in $\mathbb{C}^2$ it's just applying the matrix to the vector; if I have understood $\mathbb{C}[X,Y]$ correctly, is it the action for which the matrix $(A)_{i,j}$ sends $X \longmapsto A_{11}X + A_{12}Y$ and sends $Y \longmapsto A_{21}X+A_{22}Y$? (And then in the sum $(*)$, $X^i \longmapsto (A_{11}X + A_{12}Y)^i$, etc.)
Is that correct? If not, please let me know what I got wrong! Thanks