I wanted to construct transition functions for the tautological bundles on $\mathbb{R}P^n$ and $\mathbb{C}P^n$, but I guess I have not understood the concept of tautological bundle well enough.
Can someone clarify, and give hints for the question?
All I know is that the tautological bundle of $\mathbb{R}P^n$ is given by the projection $\pi$
$\pi: E=\{(L,v) \in \mathbb{R}P^n \times \mathbb{R}^{n+1} : v \in L \} \to \mathbb{R}P^n $.