I was studying vector bundles and didn't get the transition function's definition. Suppose we have two trivializations $\phi_U, \phi_V$. Taking the composition $\phi_u^{-1}\circ\phi_V: (U \cap V) \times \mathbb{R}^n \to (U \cap V) \times \mathbb{R}^n$ on the intersection, my textbook says that:
$$(x, v) \mapsto (x, g_{UV}(v))$$
Where $g_{UV}: (U \cap V) \to GL_n(\mathbb{R})$.
I understood why we have the identity on the first coordinate, but in my mind the second one should be the image $f(x, v)$ of a continuous function $f: (U \cap V) \times \mathbb{R}^n \to \mathbb{R}^n$, only. From cartesian closedness of $Set$ we get a unique $g_{UV}: (U \cap V) \to \{g| g: \mathbb{R}^n\to \mathbb{R}^n\}$ from $f$, but I can't see how we can assure the linear structure on the codomain. Somebody could help me in that?