dual basis

The dual basis is the set of perfectly tuned 'coordinate readers' attached to a chosen basis: each one outputs the coordinate of a vector along exactly one basis direction and ignores all the others. If your basis names the directions, the dual basis hands you the numbers you would write next to each direction.

Let V be finite-dimensional over K with basis e_1, ..., e_n. The dual basis e_1*, ..., e_n* of V* is defined by the Kronecker condition e_i*(e_j) = δ_ij, equal to 1 if i = j and 0 otherwise. These n functionals are linearly independent and span V*, so they genuinely form a basis; for any vector v = Σ c_i e_i one has e_i*(v) = c_i, the i-th coordinate.

The dual basis gives the concrete isomorphism V ≅ V* (basis-dependent, not canonical) and underlies index notation in tensor calculus: writing a tensor in coordinates means expanding it in tensor products of basis vectors e_i and dual basis covectors e_j*. Changing the basis of V changes the dual basis by the inverse-transpose matrix — the reason vectors and covectors are said to transform contravariantly versus covariantly.