In Einstein notation, why do basis vectors have lower indices?

Question

For example, the vector V can be written as $V=V^\mu\hat{e}_\mu$.

Why does the basis vector have covariant lower indices?

Answer 1

The easiest explanation is because you're supposed to add the terms of a linear combination. Since the coordinates of the vector have upper index the base has to be indexed with lower index.

You can also understand this by the fact that the base is covariant. This may be easier to see if you consider vectors as tuples instead, that is you write $V^\nu = V^{\mu}e_\mu^\nu$ (in fact the basis works as a linear map from tuples to vectors).