I am reading Mumford's 1970 Oslo lecture on moduli theory.
In this lecture, he defines a family of n-dimensional vector space endomorphisms over a scheme $T$ to be a pair $(\mathcal{E}, \phi)$ where $\mathcal{E}$ is a rank $n$ vector bundle over $T$ and $\phi$ is a vector bundle endomorphism of $\mathcal{E}$.
The proof that there exists n fine moduli space $M$ representing such families is quite short: Namely, let $(\mathcal{E}, \phi)$ be any such family over a scheme $T$, then for any non-trivial line bundle $L$ on $T$, the pair $(\mathcal{E} \otimes L, \phi \otimes 1_L)$ and $(\mathcal{E}, \phi)$ determine the same morphism $T \to M$.
Why do $(\mathcal{E}, \phi)$ and $(\mathcal{E} \otimes L, \phi \otimes 1_L)$ determine the same morphism $T \to M$?