Something's been bothering me about all of this. I think I understand now. I don't believe a basis can be constructed until a coordinate system is defined. I general, geometric objects are not dependant on a particular basis to be defined. However that can't be said for the basis vectors themselves since they are defined according to a coordinate system. E.g. the basis vectors that we all know and love i, j, k are defined as being parallel to the x, y and z axes respectively. So that person's claim that one doesn't need a coordinate system to make measurements against is false. E.g. for a Cartesian coordinate system you have to know the direction of the x, y and z axes in order to know what direction the basis vectors i, j, k are pointing.

Okay. I'm happy now.