3.4 Tensor Dual Basis Structure

Tensor Dual Basis Structure is the detailed account of the dual basis construction, {e^i} built from a basis {e_i} of V, covering its existence and uniqueness proof, its characterization as the coefficient-extraction functionals for a given basis, its behavior under change of basis, and its role as the canonical bridge that makes coordinate computation with covectors as concrete and mechanical as coordinate computation with ordinary vectors.

Existence and Uniqueness of the Dual Basis

Construction

Given a basis {e_1, ..., e_n} of V, define e^i : V → F on basis vectors by e^i(e_j) = δ^i_j and extend linearly, using the fact, established in tensor covector linearity structure, that a linear functional is uniquely determined by its values on a basis. This defines e^i unambiguously for each i, since the extension formula e^i(Σ_j v^j e_j) = Σ_j v^j δ^i_j = v^i involves no unresolved choices.

Linear Independence and Spanning

The set {e^1, ..., e^n} is linearly independent: if Σ_i c_i e^i = 0, evaluating at e_j gives Σ_i c_i δ^i_j = c_j = 0 for every j, so all coefficients vanish. It spans V*: for any ω ∈ V*, the functional Σ_i ω(e_i) e^i agrees with ω on every basis vector e_j, hence agrees with ω everywhere by the uniqueness-from-basis-values principle, showing ω = Σ_i ω(e_i) e^i. Together, independence and spanning confirm {e^i} is a basis of V*.

Coefficient-Extraction Characterization

e^i as a Coordinate Reader

The dual basis vector e^i is exactly the functional that reads off the i-th coordinate of any vector relative to {e_i}: for v = Σ_j v^j e_j, e^i(v) = v^i. This characterization is not merely a consequence of the defining formula but is, in practice, the most useful way to think about what a dual basis vector does: it is a coordinate-extraction device tied to one specific basis vector of V.

Dependence on the Original Basis

Because e^i(v) = v^i depends on the coordinates of v relative to {e_j}, changing the original basis of V changes the meaning of e^i correspondingly; the dual basis is not an intrinsic structure on V* alone but is defined only relative to a prior choice of basis on V, consistent with the absence of a natural isomorphism V ≅ V*.

Change of Basis for the Dual Basis

Transformation Rule

If e'_i = Σ_j a^j_i e_j, the corresponding dual basis {e'^i} satisfies e'^i = Σ_j b^i_j e^j, using the inverse matrix B = A^{-1}. This can be verified directly: e'^i(e'_k) = Σ_j b^i_j e^j(Σ_l a^l_k e_l) = Σ_{j,l} b^i_j a^l_k δ^j_l = Σ_j b^i_j a^j_k = δ^i_k, the last equality because BA = I.

The Inverse-Transpose Relationship

The dual basis transforms by the inverse of the matrix transforming the original basis, in contrast to vector components, which transform by the inverse alone applied to components rather than to basis vectors; tracking carefully which of the four related objects, original basis, dual basis, vector components, covector components, transforms by A, A^{-1}, or their transposes is essential to using dual basis structure correctly, and is precisely the content of the covariant/contravariant distinction established throughout dual spaces and covectors and tensor isomorphism coordinate transfer.

Computational Role of the Dual Basis

Recovering Coefficients in Any Expansion

Given any basis {e_i} of V and a vector v, the dual basis supplies the direct formula v = Σ_i e^i(v) e_i for recovering v from its coordinates, without needing to solve a linear system explicitly; the coordinates v^i = e^i(v) are computed by direct evaluation once the dual basis has been constructed.

Dual Basis in Matrix Terms

If the basis {e_i} is written as the columns of an invertible matrix M, the dual basis functionals {e^i} correspond to the rows of M^{-1}, since e^i(e_j) is the (i, j) entry of M^{-1} M = I; this gives a fully computational route to the dual basis, reducing its construction to a single matrix inversion once the original basis vectors are expressed in coordinates.