Let $M$ be a n-manifold and $(E,\pi,M)$ a bundle over $M$ with fiber $E_{m}, m \in M$; let $(\phi_{\alpha,\beta})$ the transition maps and $(U_{\alpha})$ an open cover of $M$.
Then I define $(\Lambda^{k}E)_{m}=\Lambda^{k}E_{m}$ and I want to build the exterior bundle. I cannot understand what the lectures did:
He defined $$\Lambda^{k}\phi_{\alpha,\beta}(m): \Lambda^{k}E_{m} \rightarrow \Lambda^{k}E_{m}$$ as $$\Lambda^{k}(\phi_{\alpha,\beta})(\theta)(v_{1},...v_{k})=\theta(\phi_{\alpha,\beta}(m)v_{1},...,\phi_{\alpha,\beta}(m)v_{k})$$ and he then noticed that this is the family of the transition maps for the exterior bundle, i.e., $\cup_{m \in M} \Lambda^{k}E_{m}$. But I studied that the transition maps should be defined on $U_{\alpha} \cap U_{\beta}$, so what I am missing?