3.4.1 Tensor Dual Basis Construction

Tensor Dual Basis Construction is the concrete, step-by-step procedure for producing the dual basis functionals {e^i} from an explicit basis {e_i} given in coordinates, covering the reduction of the construction to a single matrix inversion, a worked numerical example carrying the procedure through completely, and the alternative, row-by-row method of solving a separate linear system for each dual basis vector when only the defining conditions e^i(e_j) = δ^i_j are used directly.

Reduction to a Matrix Inversion

Setting Up the Matrix

Suppose V = F^n with its standard coordinates, and {e_1, ..., e_n} is a basis given as column vectors, assembled into an invertible matrix M whose j-th column is e_j. Because e^i(e_j) = δ^i_j, writing each e^i as a row vector r_i acting on column vectors by matrix multiplication, the condition e^i(e_j) = δ^i_j becomes exactly the statement that the matrix R with rows r_1, ..., r_n satisfies RM = I, so:

The rows of M^{-1} are exactly the dual basis functionals, expressed as row vectors ready to act on column-vector inputs by matrix multiplication.

Worked Example in F^2

The Given Basis

Take V = F^2 and the basis e_1 = (2, 1), e_2 = (1, 1), assembled as:

Computing the Inverse

det(M) = 2(1) - 1(1) = 1, so M^{-1} = (1/1) [[1, -1], [-1, 2]] = [[1, -1], [-1, 2]]. The rows give e^1 = (1, -1) and e^2 = (-1, 2), meaning e^1(v) = v_x - v_y and e^2(v) = -v_x + 2v_y for v = (v_x, v_y).

Verification

Checking directly: e^1(e_1) = 2 - 1 = 1, e^1(e_2) = 1 - 1 = 0, e^2(e_1) = -2 + 2 = 0, e^2(e_2) = -1 + 2 = 1, matching e^i(e_j) = δ^i_j exactly, confirming the construction.

Row-by-Row Construction via Individual Linear Systems

Solving for One Dual Basis Vector at a Time

An alternative to inverting M all at once is to solve, for each i, the linear system Σ_j r^{(i)}_j (e_j)_k = δ^i_k for the unknown coefficients r^{(i)}_j defining e^i in terms of the standard dual basis, treating each e^i as a separate unknown row vector satisfying n scalar equations, one for each basis vector e_j it must be tested against.

Equivalence to the Matrix Method

This row-by-row approach solves the same underlying linear system as the full matrix inversion, just one row of R = M^{-1} at a time rather than all rows simultaneously; for hand computation with a specific single dual basis vector needed, solving one such system directly can be more efficient than computing the full inverse, while for producing the entire dual basis at once, the matrix-inversion method is generally preferred.

Construction Relative to a Non-Coordinate Basis

Using an Auxiliary Coordinate System

When V is an abstract vector space without a pre-given coordinate system, the dual basis can still be constructed by first choosing any convenient reference basis to assign coordinates to {e_i}, carrying out the matrix-inversion procedure in those coordinates, and then reinterpreting the resulting row vectors as abstract functionals via the same reference basis; the numerical procedure is identical, only the bookkeeping of translating between the abstract space and its coordinate representation changes.

Basis Independence of the Final Result

Although the intermediate computation depends on the auxiliary reference coordinates chosen, the resulting dual basis {e^i} is uniquely determined by the original basis {e_i} alone, matching the uniqueness already established in tensor dual basis structure; different choices of auxiliary coordinates used purely for computational convenience necessarily produce the identical final functionals e^i, since the defining conditions e^i(e_j) = δ^i_j pin down each e^i completely regardless of how the computation is carried out.