9.15.1 Tensor Dual Basis Evaluation Pair

The Tensor Dual Basis Evaluation Pair connects dual bases through bilinear forms, revealing structure in tensor algebra.

Tensor Dual Basis Evaluation Pair is the specific combination of one dual basis covector and one basis vector taken together as the unit that must be evaluated to test or exploit the duality relationship between a basis and its dual. It refers to the pair itself, as an object of evaluation, rather than to the numerical outcome of that evaluation.

Structure of the Pair

Two Complementary Elements

An evaluation pair always consists of one element drawn from the dual basis, a covector carrying an upper index, and one element drawn from the primary basis, a vector carrying a lower index. Evaluating the pair means applying the covector to the vector as a linear functional acting on its argument.

(e^{i}, e_{j}) \mapsto e^{i} (e_{j})

Every Possible Pair in a Basis

For a basis of dimension n, there are n squared distinct evaluation pairs, one for every combination of a dual basis covector and a basis vector, since each of the n dual basis covectors can be paired with each of the n basis vectors.

Outcome of Evaluating the Pair

Matching Index Pairs

When the index of the dual basis covector matches the index of the basis vector, evaluating the pair yields the value one, reflecting the defining property that a dual basis covector acts as the identity on its own corresponding direction.

Non-Matching Index Pairs

When the indices differ, evaluating the pair yields the value zero, reflecting that a dual basis covector is constructed to vanish on every basis direction other than its own.

\sum_{i, j} e^{i} (e_{j}) = \sum_{i, j} δ_{j}^{i}

Use of Evaluation Pairs

Building the Full Duality Table

Collecting the outcomes of every evaluation pair in a basis produces a complete table of ones and zeros indexed by the two basis indices, and this table is precisely the Kronecker delta viewed as a matrix, summarizing the full duality relationship between the basis and its dual in one structure.

Testing Candidate Dual Bases

Given a proposed set of covectors intended to serve as a dual basis, evaluating every pair formed with the primary basis vectors and checking that the resulting table matches the expected pattern of ones on matching indices and zeros elsewhere is the direct procedure for confirming that the proposed covectors genuinely form the dual basis.

Extracting Individual Coordinates

Evaluating the pair formed from a specific dual basis covector and an arbitrary vector, expanded in the primary basis, isolates the coordinate of that vector associated with the covector's index, since every other term in the expansion evaluates to zero against that covector.

Behavior Under Basis Change

Pairs Must Be Evaluated Within a Single Basis

An evaluation pair is only meaningful when both its covector and its vector belong to the same coordinated basis and dual basis. Mixing a dual basis covector from one basis with a basis vector from a different, untransformed basis does not correspond to a valid evaluation pair and does not yield the expected Kronecker delta pattern.

Consistency Preserved Across Transformation

When a basis and its dual are transformed together using the standard basis change rule, every evaluation pair formed from the new dual basis and new basis vectors reproduces the same pattern of ones and zeros as the original pairs, confirming that the pairing structure is preserved under any valid change of basis.