(Unrelated to sudgy). A few years ago, I made a post here complaining about the way coordinate vectors are introduced in linear algebra. In particular, no-one ever explained me whether, if you write (2,3,4), what you mean is "the element (2,3,4) \in \R^3" or "the vector 2*b_1 + 3*b_2 + 4*b_3 for some basis {b_1, b_2, b_3}".

I saw your first post while catching up this thread and I'm glad you sorted it eventually. This is taught badly all over the place and, as you've said, doing it right isn't hard!

One thing that might help as you go on is to keep in mind the distinction between an n-dimensional vector space over F, of which there are many, and the particular vector space F^n, which is just one very special example. For a general vector space, choosing a basis to write coordinates with respect to is the same as fixing an isomorphism with F^n. (If the basis is v_1, ..., v_n then the isomorphism is just (x_1, ..., x_n) |-> x_1v_1 + ... + x_nv_n.)

(And eventually you'll want to avoid choosing a basis/using coordinates if at all possible!)