Let $ n \in \mathbb{N}$ be an integer ,$ n\geq 1$ and let $ M = P ( \mathbb{R}^{n+1})$ denote the real projective space of dimension n. Let $(x_0,...,x_n)$ the coordinates on $\mathbb{R}^{n+1}$ we use the $n+1$ we use the $n+1$ coordinate charts $\psi_i : O_i \subseteq \mathbb{ R}^n \to U_i$ given by
$\psi_i ( x_0,...,x_{i-1} , x_{i+1} ,...,x_n) = \mathbb{R} ( x_0,..., x_{i-1} ,1 , x_{i+1} , ...,x_n)$.
Compute the change of coordinates functions $\psi^{-1}_i \circ \psi_j $ for $i<j$.
$\psi_{i}^{-1} ( r x_0,..., rx_{i-1} ,r, rx_{i+1} ,..., rx_n) = ( x_0,...,x_{i-1}, x_{i+1},...,x_n)$.
But by this inverse formula: $\psi^{-1}_i \circ \psi_j $ doesn't make sense.
Can you please help me compute this change of coordinates functions? This type of questions of computing change of coordinate functions occurs many times.