Hartshorne defines a geometric vector bundle as follows -
Let $Y$ be a scheme. A (geometric) vector bundle of rank $n$ over $Y$ is a scheme $X$ and a morphism $f:X\rightarrow Y$ together with additional data consisting of an open covering $\{U_i\}$ of $Y$ and isomorphisms $\psi_i:f^{-1}(U_i)\rightarrow \mathbb A^{n}_{U_i}:=\text{ Spec }\mathbb Z[x_1,\cdots,x_n]\times_\mathbb Z U_i$ such that for any $i,j$ and for any open affine subset $V=\text{ Spec }A\subseteq U_i\cap U_j$ the automorphism $\psi = \psi_j\circ\psi_{i}^{-1}$ of $\mathbb A^{n}_{V} = \text{Spec }A[x_1,\cdots,x_n]$ is given by a linear automorphism $\theta$ of $A[x_1,\cdots,x_n]$, i.e, $\theta(a)=a$ for any $a\in A$ and $\theta(x_i)=\sum a_{ij}x_j$ for suitable $a_{ij}\in A$
I have the following question -
Is it clear just from definition that if $U\subseteq U_i$ for some $i$ then $\psi_i(f^{-1}(U))=\mathbb A^n_U$?