3.9.5 Tensor Dual Coordinate Reconstruction

Tensor Dual Coordinate Reconstruction is the reverse procedure to coordinate assignment: given an ordered list of n numbers relative to a chosen basis, reconstruction builds the unique covector in V* whose coordinates are exactly that list, completing the two-way correspondence between abstract covectors and numerical component data. Where assignment moves from a covector to its coordinates, reconstruction moves from coordinates back to a covector, and the two procedures are mutually inverse, forming a full isomorphism between V* and F^n relative to a fixed basis.

The Reconstruction Formula

Building a Covector from a Coordinate List

Given a coordinate list (c_1, c_2, ..., c_n) in F^n and a basis e_1, ..., e_n of V with dual basis e^1, ..., e^n, the reconstructed covector is defined as the linear combination

using the Einstein summation convention, f = c_i e^i. This expression is a well-defined element of V*, since V* is a vector space and any linear combination of its elements, here the dual basis covectors, remains within V*.

Verifying the Reconstruction Matches the Original List

To confirm the reconstruction is correct, evaluate f on each basis vector e_j:

so the coordinate assignment applied back to f returns exactly the original list (c_1, ..., c_n), confirming that reconstruction is a genuine right inverse to the assignment procedure.

Reconstruction and Assignment Are Mutually Inverse

Reconstruction After Assignment

Starting instead from an arbitrary covector g, applying assignment produces its coordinate list (g_1, ..., g_n) with g_i = g(e_i), and applying reconstruction to this list produces g_i e^i. Since every covector decomposes uniquely as a linear combination of dual basis elements with its own coordinates as coefficients, g_i e^i = g exactly, confirming reconstruction is also a left inverse.

The Full Isomorphism

Together, these two verifications establish that reconstruction and assignment are inverse bijections between V* and F^n, and this pair of maps constitutes the concrete isomorphism V* ≅ F^n that depends on, and is entirely determined by, the choice of basis used to build the dual basis.

Reconstruction from Partial or Alternative Data

Reconstructing from Action on Non-Basis Vectors

Reconstruction as described requires the coordinate list relative to the dual basis specifically. If instead a covector's values are known on a set of vectors that do not form a basis, or are known incompletely, reconstruction is not directly possible from the formula f = c_i e^i; the values on a full basis, or an equivalent amount of independent linear information, are required to pin down f uniquely.

Reconstruction Under a Change of Basis

If a coordinate list is given relative to one basis but a covector is desired described relative to another, reconstruction should be preceded by applying the coordinate transformation rule to convert the list to the new basis, since reconstructing directly with the wrong dual basis produces a different covector than intended.

Numerical Example

Reconstructing a Covector on R^3

Given the coordinate list (4, -2, 1) relative to the standard basis of R^3 and its dual basis e^1(x, y, z) = x, e^2(x, y, z) = y, e^3(x, y, z) = z, the reconstructed covector is

which evaluated on a general vector (x, y, z) gives f(x, y, z) = 4x - 2y + z, and substituting the basis vectors themselves recovers exactly the original coordinates 4, -2, 1.

Practical Significance

Turning Data Back into Usable Functionals

Reconstruction is essential whenever a covector arises from numerical data, such as coefficients solved from a system of equations, gradient values sampled at a point, or output from a computational procedure, since it converts that raw data back into an abstract linear functional that can be applied to any vector in V, not merely to the specific basis vectors used to compute it.

Foundation for Working with General Tensors

The reconstruction procedure for covectors generalizes directly to reconstructing higher-rank tensors from their full array of components: a (p, q) tensor is rebuilt from its component array by summing the appropriate tensor products of basis vectors and dual basis covectors weighted by each component, exactly mirroring the single-index reconstruction described here.

Diagrammatic Summary

The diagram shows reconstruction and assignment as a mutually inverse pair of maps completing the identification between coordinate lists in F^n and covectors in V*.