I'm curious as to what are all the line bundles on $\mathbb{P}^2 \times \mathbb{P}^2$? I might be mistaken but I believe they are classified as $$ \mathcal{O}_{\mathbb{P}^2 \times \mathbb{P}^2}(p,q) := \pi_1^* \mathcal{O}_{\mathbb{P}^2}(p)\otimes \pi_2^* \mathcal{O}_{\mathbb{P}^2}(q) $$ where $\pi_i: \mathbb{P}^2 \times \mathbb{P}^2 \rightarrow \mathbb{P}^2$ is the map onto the i-th factor. Is this correct? If so, then why is this true? Is it true for just line bundles or any rank vector bundle?
Also, if I have a hypersurface of this space, say, $H$, then are the line bundles on $H$ given by $$ \mathcal{O}_{H}(p,q):= \mathcal{O}_{\mathbb{P}^2 \times \mathbb{P}^2}(p,q)\vert_{H} \quad ? $$
I believe this is true as well and has something to do with the Lefshetz Hyperplane Theorem but I'm just not seeing how to put the pieces together.
Oh, and everything is complex.
Thanks for the help.