How to prove that a vector bundle is trivial iff there are n global sections that form a basis on each fiber?
Solution 1:
The trick is to work with a local trivialization of $E$ since smoothness of a map is a local property. Now, for both parts 1 and 2 use the fact that inverse of an invertible matrix $A$ depends smoothly on its entries and hence, solution vector of a linear system $Ax=b$ depends smoothly on $A$ and $b$.