Cobases

Cobases are dual collections associated with a basis of a vector space. In a finite-dimensional vector space V over a field F, a basis B = {v1, ..., vn} has a corresponding dual basis B* = {v^1, ..., v^n} in the dual space V*, defined by v^i(v_j) = δ_j^i (the Kronecker delta). The cobase consists of linear functionals that extract the coordinates of vectors with respect to B: if v = sum_i a_i v_i, then the coordinates a_i are given by a_i = v^i(v).

These dual bases are used to represent linear maps and coordinate frames, and to raise and lower indices in tensor calculus when a bilinear form, such as a metric, is present. The pairings v^i(v_j) = δ_j^i encode a perfect pairing between V and V*. There is a natural isomorphism between V and V** via v ↦ φ ↦ φ(v).

Example: In R^3 with the standard basis e1, e2, e3, the dual basis is e^1, e^2, e^3, satisfying e^i(e_j) = δ_ij; the covectors e^i correspond to the coordinate projections onto the i-th component. In general, changing bases induces a corresponding transformation of the cobase to maintain consistent coordinates and functional representations.

Notes: The term "cobase" is sometimes used interchangeably with "dual basis" in older literature; however "dual basis" is the standard term in modern linear algebra. Some usages in other branches (or topology) may refer to different concepts, such as subbases, so context is important.