Surely contravariant vectors and covariant ones live in different vector spaces? That is, if the vector space containing the contravariant vectors is V, then the covariant vectors live in the dual space V*: they are really linear maps taking the vectors in V to the underlying field of scalars. There is symmetry because the dual space V** of the dual space V* is isomorphic (in a canonical way not depending on a choice of basis) to the original space V, so that V may be naturally considered the dual space of V* as well as V* being the dual space of V. But there is no such canonical isomorphism between V and V*. The upshot is that you cannot equate a vector in the original space with a vector in the dual space as you do at 2.20.
I've always wondered: If V is finite-dimensional with a basis B, why isn't the isomorphism to V* with a basis B* (such that b*_i(b_j) = δ_ij) considered canonical?
@@APaleDot Just because canonical *means* independent of a choice of basis. eg consider V = R² over the scalar field R. One basis is e_1 = (1,0), e_2 = (0,1), and then V*, the space of linear maps from V to R, has the dual basis E_1, E_2 defined by E_i(e_j) = δ_ij exactly as you say. And then K: e_i -> E_i is an isomorphism between V and V*, but not a canonical one. For example we can define another basis (f_i) for V by f_1 = (1,1), f_2 = (-1,1). Then eg f_1 = e_1 + e_2 while e_1 = 0.5f_1 - 0.5f_2. The new dual basis in V* corresponding to the f_i in V will be F_i defined by F_i(f_j) = δ_ij and this induces another isomorphism between V and V*, namely that given by L:f_i -> F_i. These isomorphisms (K and L) are not the same. Notice for example that the latter isomorphism L maps eg e_1 (= 0.5f_1 - 0.5f_2) to 0.5F_1 - 0.5F_2 (by linearity), and (again using linearity) this linear map (from V to R) maps e_1 to 0.5F_1(e_1) - 0.5F_2(e_1) = 0.5F_1(0.5f_1 - 0.5f_2) - 0.5F_2(0.5f_1 - 0.5f_2) = 0.25δ_11 - 0.25δ_12 - 0.25δ_21 + 0.25δ_22 = 0.25 + 0.25 = 0.5 and similarly maps e_2 to 0.5F_1(e_2) - 0.5F_2(e_2) = 0.5F_1(0.5f_1 + 0.5f_2) - 0.5F_2(0.5f_1 + 0.5f_2) = 0.25δ_11 + 0.25δ_12 - 0.25δ_21 - 0.25δ_22 = 0.25 - 0.25 = 0 In other words, while K(e_1) = E_1, L(e_1) = 0.5E_1 so the two isomorphisms K and L from V to V* are certainly not the same. All this illustrates that the basis we choose for V determines the induced isomorphism mapping V to V*.
