3.9.1 Tensor Dual Coordinate Assignment

Tensor Dual Coordinate Assignment is the specific procedure that assigns to every covector in V* a unique ordered list of n numbers relative to a chosen basis of V, realizing the abstract dual space as the concrete coordinate space F^n once that basis has been fixed. This assignment is a linear isomorphism V* -> F^n, and understanding it as an assignment procedure, rather than only as an abstract fact, clarifies exactly how any given covector is turned into usable numerical data.

The Assignment Map

Definition

Fix a basis e_1, ..., e_n of V with dual basis e^1, ..., e^n. The dual coordinate assignment is the map

which assigns to each covector f the ordered tuple of its values on the basis vectors, that is, precisely its coordinate description (f_1, ..., f_n).

Well-Definedness

The assignment Φ is well-defined because a linear functional is completely determined by its values on a basis: once f(e_1), ..., f(e_n) are known, the value of f at any other vector v = v^i e_i is forced to equal f_i v^i by linearity, so no ambiguity remains in the assignment.

Linearity and Bijectivity of the Assignment

Linearity

The assignment map is linear, since for covectors f, g and scalars a, b,

because evaluating af + bg at any basis vector e_i gives a f(e_i) + b g(e_i) directly from the pointwise definition of covector addition and scaling.

Injectivity

The assignment is injective: if Φ(f) = Φ(g), then f and g agree on every basis vector, and since a linear functional is determined by its action on a basis, f = g. No two distinct covectors are ever assigned the same coordinate tuple.

Surjectivity

The assignment is surjective: given any tuple (c_1, ..., c_n) in F^n, defining f = c_i e^i produces a covector whose coordinate assignment is exactly (c_1, ..., c_n), since f(e_j) = c_i e^i(e_j) = c_i δ^i_j = c_j. Every possible coordinate tuple is attained by some covector.

Conclusion: An Isomorphism

Being linear, injective, and surjective, the assignment Φ is a linear isomorphism between V* and F^n, confirming that once a basis of V is fixed, the dual space can be identified completely and faithfully with ordinary coordinate tuples.

Assignment Depends on the Chosen Basis

Different Bases Give Different Assignments

The specific numbers assigned to a covector f depend entirely on which basis of V was used to build the dual basis. Changing the basis of V changes the assignment map Φ, and the same abstract covector f receives a different coordinate tuple relative to the new basis, related to the old tuple by the covariant transformation rule.

The Assignment Is Not Canonical

Unlike the natural pairing itself, which is basis-free, the coordinate assignment Φ is explicitly basis-dependent by construction, since it relies on a chosen basis to define the dual basis used for extraction. This is an intentional and necessary feature: coordinates are, by their nature, a basis-relative concept, distinct from the coordinate-free covector they describe.

Practical Use of the Assignment

Turning Abstract Covectors into Computable Data

The primary practical value of the assignment procedure is converting an abstractly defined linear functional, perhaps specified as a formula or a geometric condition, into an explicit list of numbers that can be stored, manipulated, and combined using ordinary arithmetic and linear algebra software.

Reassembling a Covector from Its Assignment

Given an assigned coordinate tuple (f_1, ..., f_n), the original covector is reconstructed as f = f_i e^i, and its value at any vector v = v^i e_i is recovered by the summation formula f(v) = f_i v^i, completing the round trip between the abstract object and its numerical description.

Diagrammatic Summary

The diagram shows the assignment map Φ and its inverse establishing a two-way, faithful correspondence between covectors in V* and coordinate tuples in F^n.