3.4.4 Tensor Dual Basis Coordinate Extraction

Tensor Dual Basis Coordinate Extraction is the general technique of recovering the coordinate array of any tensor, of any variance type and any rank, by evaluating the tensor directly on the appropriate combination of basis vectors and dual basis vectors, generalizing the extraction of a single vector's coordinates via v^i = e^i(v) and a single covector's coordinates via ω_i = ω(e_i) to the fully general (p, q)-tensor case.

Extracting Coordinates of a Vector and a Covector

The Two Elementary Cases

A vector's coordinates are extracted by applying dual basis vectors, v^i = e^i(v), since e^i reads off exactly the i-th coefficient in the expansion v = Σ_j v^j e_j. A covector's coordinates are extracted by applying it to basis vectors, ω_i = ω(e_i), since evaluating ω = Σ_k ω_k e^k at e_i gives Σ_k ω_k δ^k_i = ω_i. These two extraction rules are structurally dual to one another: a vector's coordinates come from applying dual basis covectors to it, and a covector's coordinates come from applying it to basis vectors.

General Extraction for a (p, q)-Tensor

The Extraction Formula

For a (p, q)-tensor T, regarded as a multilinear functional accepting p covector arguments and q vector arguments, the coordinate array T^{i_1...i_p}_{j_1...j_q} relative to a basis {e_i} and its dual {e^i} is extracted by evaluating T on the dual basis vectors in the upper argument slots and the basis vectors in the lower argument slots:

This is the direct generalization of the two elementary extraction rules: upper indices of T are read by supplying dual basis covectors e^{i_k} to the covector-accepting slots, and lower indices are read by supplying basis vectors e_{j_l} to the vector-accepting slots.

Why the Formula Works

The formula follows by linearity from the fact that any simple term v_1 ⊗ ⋯ ⊗ v_p ⊗ ω_1 ⊗ ⋯ ⊗ ω_q of T, when evaluated on e^{i_1}, ..., e^{i_p}, e_{j_1}, ..., e_{j_q}, gives ∏_k e^{i_k}(v_k) · ∏_l ω_l(e_{j_l}), and each factor is exactly the coordinate-extraction value already established for a single vector or covector; the general extraction formula is simply this elementary extraction applied independently, slot by slot, to a simple tensor, then extended by linearity to general tensors.

Extraction for the Matrix of a Linear Map

Recovering Matrix Entries

For a linear map φ ∈ Hom(V, W) ≅ V* ⊗ W, identified as a (1, 1)-tensor with one upper index from W and one lower index from V, the extraction formula specializes to a^j_i = f^j(φ(e_i)), using a basis {e_i} of V and the dual basis {f^j} of W*; this recovers the familiar procedure of finding a matrix's j-th row, i-th column entry by applying the map to the i-th basis vector of the domain and reading off the j-th coordinate of the result.

Consistency With Direct Matrix Computation

Because φ(e_i) = Σ_j a^j_i f_j by definition of the matrix representation, applying f^j gives f^j(φ(e_i)) = Σ_k a^k_i f^j(f_k) = Σ_k a^k_i δ^j_k = a^j_i, exactly recovering the entry a^j_i; the extraction formula and the ordinary definition of a matrix representation agree completely, confirming that dual basis coordinate extraction is not a separate technique from standard matrix bookkeeping but its precise, index-explicit formulation.

Practical Use in Verifying Tensor Identities

Checking Equality Componentwise

Two tensors T and S of the same type are equal if and only if their extracted coordinate arrays agree in every slot, T^{i_1...i_p}_{j_1...j_q} = S^{i_1...i_p}_{j_1...j_q} for every choice of indices; because coordinate extraction reduces this check to evaluating both tensors on the same finite list of basis and dual basis arguments, it supplies a completely mechanical procedure for verifying claimed tensor identities, reducing an abstract equality of multilinear functionals to a finite list of numerical or symbolic evaluations.

Basis Independence of the Underlying Claim

Although the extraction procedure itself depends on a chosen basis, the resulting statement that two tensors are equal is a basis-independent fact once verified in any single basis, since a tensor equality holding in one basis, by the coordinate transfer rules described in tensor isomorphism coordinate transfer, automatically holds in every other basis as well.