I'm currently reading "Tensors, Differential forms and Variational Principles" By D. Lovelock and H. Rund. This is from page 2 :
and then on page 7, concerning equation 1.1
I fail to understand why equation 1.1 should depend on a coordinate system. I seems to me that the $\mathbf{e}_i$, as defined in the first paragraph, are legit vectors independently of them being used as a basis for the space.
Are the authors implying that the symbols $\mathbf{e}_i$ are just placeholders, whose actual vectors they are referring to would change depending on the choice of coordinate system ?