Let $X$ be a variety and $\varphi : F_1 \to F_2$ be a morphism of vector bundles over $X$. Then it is easy to check that the locus on $X$ for which $\varphi$ vanishes is a closed subscheme of $X$. Furthermore, the ideal sheaf cutting out this locus can be described locally by trivializing both vector bundles.

Let us try to do this in families now.

Take a family of varieties $\pi:\mathcal{X} \to B$ over some base scheme $B$. We will not assume $B$ is reduced but we can assume the usual nice things about $\pi$. Lets say $\pi$ is flat, proper and of finite presentation with *nice* fibers. Take a morphism of vector bundles $\varphi: F_1 \to F_2$ on $\mathcal{X}$.

We want to solve the following:

Vaguely:What is the locus in $B$ like over which $\varphi$ is precisely zero.

Observe that this is the same situation as in the very first sentence of this post, if we take $\pi = \operatorname{id}_{\mathcal{X}}$. But in our current generality we need to make things a little more precise for the question to make sense.

Define the functor $G \subset \hom(\_,B)$ so that $T\to B$ is in $G$ iff $\varphi|_{\mathcal{X}_T}$ is the zero morphism.

Precisely:Is $G$ representable? If it is a closed subscheme, what is the ideal defining it?