Problem 1. Let π : E → X be a vector bundle, and let s1, s2 : X → E be the two smooth sections of E. We define their sum s1 + s2 by formula: (s1 + s2)(x) := s1(x) + s2(x), x ∈ X.
1) Show that s1 + s2 is a smooth section of E.
2) Let Γ(E) = C∞(E) be the set of all smooth sections of E. Define the multiplication of sections from Γ(E) by real numbers in a natural way. Is Γ(E) an R-linear space?
Problem 2. How to define a structure of C∞(X)-module on Γ(E)?
Problem 3. Let π1 : E1 → X and π2 : E2 → X be two smooth vector bundles over X. How to define their direct sum and their tensor product as smooth vector bundles.
Problem 4. Suppose there are given transition functions {φij} and {ψij} for the bundles π1 : E1 → X and π2 : E2 → X. What are the transition functions for the vector bundles E1 ⊕ E2, E1 ⊗ E2, E ͮ1.

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