I am reading Kunz's "Introduction to Commutative Algebra and Algebraic Geometry" and I'm not understanding what is meant by the underlined sentence in the screenshot below. What exactly is invariant here? If $T: \mathbb{P}^n \to \mathbb{P}^n$ is the corresponding coordinate transformation, do we have $T(V) = V$ for any projective variety $V \subset \mathbb{P}^n$? I don't think so because if I take $V = V(x^2 + y^2 -xy) \subset \mathbb{P}^2(\mathbb{C})$, and the transformation $T$ induced by the linear transformation given by the matrix:$ A = \begin{bmatrix} 1 & 2 \\ 2 & 1 \\ \end{bmatrix} $, if I didn't make a mistake, this is precisely a projective variety and a coordinate transformation with $T(V) \neq V$.
I appreciate any help and thank you all in advance :)