I am currently reading a book on Algebraic Geometry and both the Tautological Line Bundle $L \to \mathbb{P}_k^n$ and the blow up ${\operatorname{Bl} _0}\mathbb{A}_k^{n + 1}$ are defined set theoretically as $\left\{ {\left( {x,\ell } \right) \in \mathbb{A}_k^{n + 1} \times \mathbb{P}_k^n\;|\;x \in \ell } \right\}$.
They are defined in separate parts of the same chapter but the equivalence is never discussed. Is there a reason why the definitions coincide?