I've been studying projective space in algebraic geometry for a few days from Perrin's 'Algebraic Geometry: An introduction'. In the first page of chapter III it reads
...projective space $P^n$ contains open sets $U_i=D^+(X_i)$ which are isomorphic to affine spaces.
I know that fixed a hyperplane $\bar H\subset P^n$ one can describe $P^n$ as the disjoint union of $\bar H$ and the affine space $k^n$. To describe this correspondence I take a (!) orthogonal vector $x\in k^n$ and define $\varphi(\bar y):=(x+H)\cap \bar y$, for $\bar y\in P^n\setminus\bar H$.
My question is, in what sense does Perrin talk about isomorphism? Even, what would linear mean to talk about linear maps? I've looking for a while on the web to get an answer to this and then I found this book ('Computational line geometry', H. Pottmann and J. Wallner) that said
...attributes like unith length and orthogonality do not belong to affine geometry.
This made me wonder if my construction of $\varphi$ really makes sense. It feels like it should be an orthogonality concept in $k^n$ (cause one can multiply and sum coordinates) but that made me doubt, are there? If not, there are no better way to define $\varphi$ than picking some $x$ not in $H$?
Thank u in advance, really appreciated!
You don't need an "orthogonal" vector -- in fact, since you're working over a vector space and not an inner product space, there isn't even such a thing. You just need any vector that doesn't lie in $H$.
If you're having trouble visualizing this, notice that both $$\{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}$$ and $$\{(1, 0, 0), (0, 1, 0), (4, 7, 6)\}$$ are perfectly good bases for $\mathbb{R}^3$.
The isomorphism between $D^+(X_i)$ and $\mathbb{A}^n$ is an isomorphism of locally ringed spaces. It looks like the thing you're reading is in the introduction to the chapter where Perrin introduces those notions, so you'll have to read on to see what it means.
Any references to linear maps are talking about linear maps on the underlying vector spaces.