Suppose you have a differentiable manifold where at each point you have attached a set of basis vectors X_1,X_2,...,X_n. One thing that I don't have clear is the difference between a coordinate basis and a non-coordiante basis. I've been told that there is a way to check if the set of basis vectors is coordinate independent. One do this by simply taking the commutator between any pair of basis vectors and if the commutator is zero then my basis is not coordinate dependent. But why is that? Thanks in advance.