Classical projective transformation

Question

I have a little knowledge in projective geometry and I have some inquiries. What I know that the projective transformation is transformation by an $n\times n$ invertible matarix $A$ from a projective space to another; $L:\mathbb{P}^n\rightarrow \mathbb{P}^n$. Is it possible to apply the projective transformation from the real plane $\mathbb{R}^2$ to itself or to the real projective plane? and how? Can the real plane $\mathbb{R}^2$ be identified with $\mathbb{P}^2$?

Answer 1

You got the dimensions wrong. By definition, $P^n$ is the space of 1-dimensional linear subspaces of $\mathbb K^{n+1}$, and projective transformations are given by the action of invertible matrices of size $(n+1)\times(n+1)$. If you look at the case $n=2$, the projective plane $P^2$ is the set of lines through $0$ in $\mathbb R^3$. Whenever you fix a two-dimensional linear subspace in $\mathbb R^3$, the lines which are not contained in that plane (the "plane at infinity") form an affine plane (i.e. a copy of the affine space $\mathbb R^2$). General projective transformations do not preserve such a plane, those who do are exactly the affine transformations of that plane. You can view $P^2$ as the union of the affine plane $\mathbb R^2$ and a "line at infinity", which is a copy of $P^1$.