3.9 Tensor Dual Coordinate Description

Tensor Dual Coordinate Description is the systematic method for representing elements of a dual space V* as numerical component arrays relative to a chosen basis, together with the rules governing how those components are computed, how they combine, and how they change when the underlying basis of V is altered. Where a vector in V is described by contravariant components that multiply basis vectors directly, a covector in V* is described by covariant components that multiply dual basis covectors, and the two coordinate descriptions are linked but not identical in how they behave.

Constructing Coordinates for Covectors

The Dual Basis

Given a basis e_1, ..., e_n of V, the dual basis e^1, ..., e^n of V* is defined as the unique set of linear functionals satisfying

Every covector f in V* can then be written uniquely as a linear combination f = f_i e^i of these dual basis elements, and the numbers f_1, ..., f_n are the coordinate description of f relative to the chosen basis of V.

Extracting the Components

The coordinates of a given covector are recovered by applying it to each basis vector in turn:

This follows directly from applying f = f_j e^j to e_i and using the dual basis relation e^j(e_i) = δ^j_i to collapse the sum. This formula gives a concrete algorithm for computing the coordinate description of any covector once its action on the basis vectors of V is known.

Row Versus Column Representation

Covectors as Row Arrays

Because covector coordinates pair with vector coordinates through a single-index sum f_i v^i, it is conventional to display covector coordinates as a row array (f_1, f_2, ..., f_n) and vector coordinates as a column array, so that the pairing coincides exactly with the standard rule for multiplying a 1 x n matrix by an n x 1 matrix.

Consistency with Linear Map Representation

This row representation is consistent with how linear functionals are represented as 1 x n matrices in general linear algebra, since a linear functional V -> F acting on column vectors is precisely a 1 x n matrix, and its entries relative to a basis are exactly the coordinate description described here.

Transformation of Dual Coordinates Under a Change of Basis

The Transformation Rule

If the basis of V changes from e_1, ..., e_n to e'_1, ..., e'_n according to e'_j = A^i_j e_i for an invertible matrix A, the dual basis changes according to the inverse-transpose relationship, and the coordinates of a fixed covector f transform as

using the same matrix A that transforms the basis of V, in contrast to vector coordinates, which transform using A^{-1}.

Why the Terms Covariant and Contravariant Apply

This is the precise origin of the terminology: covector coordinates transform the same way, or covariantly, with the basis transformation A, while vector coordinates transform in the opposite, or contravariant, direction using A^{-1}. The dual coordinate description is called covariant specifically because of this shared-direction transformation behavior.

Dual Coordinates and Linear Equations

A Covector as a Linear Equation

A covector's coordinate description (f_1, ..., f_n) can be interpreted directly as the coefficients of a linear equation, f_1 x^1 + ... + f_n x^n = c, in the coordinates x^1, ..., x^n of a vector. Evaluating f at a vector with coordinates v^i computes the left-hand side of this equation at that particular point.

Kernels as Hyperplanes

The set of vectors with coordinates satisfying f_i v^i = 0 forms the kernel of f, a hyperplane through the origin of dimension n - 1, giving a direct geometric meaning to a covector's coordinate description as the normal data defining a hyperplane in coordinate space.

Practical Computation Example

Recovering Coordinates from Action on a Basis

Suppose V = R^3 with the standard basis, and a covector f is defined by its values f(e_1) = 2, f(e_2) = -3, f(e_3) = 0 on the basis vectors. By the extraction formula, the coordinate description of f is immediately (f_1, f_2, f_3) = (2, -3, 0), and for any vector v = (v^1, v^2, v^3), evaluation is computed as f(v) = 2 v^1 - 3 v^2 + 0 \cdot v^3.

Diagrammatic Summary

The diagram contrasts the column representation and contravariant transformation of vector coordinates with the row representation and covariant transformation of covector coordinates.