I'm trying to build up enough understanding to work through some GR on my own, but I'm horribly confused by some of the math concepts. So terribly so, that i'm not even sure how to ask my questions. Please bear with me. Lets work in a 2D plane. Assume I have a vector u which I can write out in terms of orthonormal basis vectors: u = 3e1 + 4e2. This gives it a length of 5. What if the basis isn't orthogonal? My vector acquires a different length and my notion of a dot product has to change, right? Now take away the coordinate system. What's left? My vector u is still there and can still be expressed by the expansion above, but it now makes no sense at all to talk about its length or the normality of the basis vectors, correct? If it's true that we cannot talk about lengths without a coordinate system, then doesn't that also mean that we cannot construct a reciprocal basis? Two basis e1 e2 e3 and e1 e2 e3 are said to be reciprocal if they satisfy the following condition: ei . ek = [tex]\delta_i^k[/tex] --Vector and Tensor Analysis with Applications by Borisenko and Tarapov But how can we satisfy that condition if we don't have a unit length? Does the presence of a reciprocal basis indicate that a coordinate system has been chosen (though the location of the origin may not be fixed)? Something else? If a column vector exists in the original basis, does the transpose of the vector exist in the reciprocal basis? I'm of the impression that the reciprocal basis is a map that takes the vector space into the real numbers. Thus, I assume it is absolutely required for dot products and any notion of the length of a vector in a non-orthogonal basis. Is that correct? If so, then all dot products should be of the form ei . ek or ei . ek, but I also read about dot products with both indices up or both down. What do those mean? Is the reciprocal basis the same place where all the covariant vectors 'live'? If not, how is the concept of a reciprocal basis related to the idea of covariance? I struggle with the idea of vectors being coordinate system independent because I equate the cotangent space with the reciprocal space and don't see how it can be defined without a coordinate representation. Well, I'm gonna cut myself off right here. Any help in answering any of these questions is greatly appreciated. Even just helping me to perhaps better formulate my questions would be a big plus. I have a BS in Physics but not really any background in the more abstract areas of math other than set theory, so it would be appreciated if answers didn't assume too much prior knowledge. Thanks a million!