6.16.3 Tensor Zero Two Dual Basis Product Relation

The Tensor Zero Two Dual Basis Product Relation explores how dual basis elements interact through tensor products in algebraic structures.

Tensor Zero Two Dual Basis Product Relation is the identity expressing an arbitrary type (0,2) tensor as a linear combination of elementary dual basis products e^i ⊗ e^j, formed by taking every possible ordered pair of dual basis covectors, and showing that these n² products together constitute a complete basis for the tensor product space V* ⊗ V*. This relation is the covariant counterpart of the basis product relation used for type (2,0) tensors, and it is what allows an abstract covariant tensor, such as a metric, to be expanded concretely as a sum of coefficients multiplying fixed dual basis products.

Constructing the Dual Basis Products

From a Dual Basis of V* to a Basis of V* Tensor V*

Given a basis {e_1, ..., e_n} of V and its associated dual basis {e^1, ..., e^n} of V*, defined by e^i(e_j) = δ^i_j, the dual basis product relation asserts that the collection of all ordered pairs:

e^{i} \otimes e^{j}

for i and j ranging from 1 to n, forms a basis for V* ⊗ V*. As with the contravariant case, there are n² such products, matching dim(V* ⊗ V*) = n².

Expansion of an Arbitrary Tensor

Any type (0,2) tensor T can be written uniquely as:

T = T_{i j} e^{i} \otimes e^{j}

with repeated indices summed. The components T_{ij} are precisely the coefficients of T in this dual basis product expansion, and evaluating both sides on a pair of basis vectors (e_k, e_l) reproduces the original component T_{kl}, confirming the consistency of the relation.

Why the Dual Basis Products Are Linearly Independent and Spanning

Linear Independence

The products e^i ⊗ e^j are linearly independent, since if a combination Σ c_{ij} (e^i ⊗ e^j) vanished as a bilinear form, evaluating it on the pair (e_k, e_l) would give exactly c_{kl}, forcing every coefficient to be zero. This evaluation trick, applying the candidate tensor to basis vectors, is the direct method for extracting components from an abstractly defined covariant tensor.

Spanning

Every elementary tensor φ ⊗ ψ expands using φ = φ_i e^i and ψ = ψ_j e^j:

φ \otimes ψ = φ_{i} ψ_{j} (e^{i} \otimes e^{j})

and since general covariant type (0,2) tensors are finite sums of such elementary products, the dual basis products span all of V* ⊗ V*.

Behavior of the Relation Under a Change of Basis

New Dual Basis Products from New Basis Vectors

If the basis of V changes via e'_i = A^k_i e_k, the dual basis transforms with the inverse matrix, e'^i = B^i_k e^k, so the new dual basis products become:

e^{' i} \otimes e^{' j} = B_{k}^{i} B_{l}^{j} (e^{k} \otimes e^{l})

Consistency with Component Transformation

Since the tensor T itself does not change, the coefficients in this new expansion must transform oppositely to the dual basis products, which is exactly the origin of the covariant transformation law T'_{ij} = A^k_i A^l_j T_{kl} for the components. This mirrors precisely, with A and B exchanged, the relationship established for the contravariant basis product relation of type (2,0) tensors.

Diagram of the Dual Product Basis Construction

Consequences of the Dual Basis Product Relation

Extracting Components by Evaluation

The dual basis product relation shows that the components of any type (0,2) tensor can always be recovered by direct evaluation on pairs of basis vectors, T_{ij} = T(e_i, e_j), providing a practical method for computing the components of an abstractly defined bilinear form, such as an inner product specified by a geometric rule, without needing to manipulate the tensor product formalism directly.

Foundation for the Metric's Component Matrix

Applying this relation to the metric tensor explains precisely why the familiar matrix of a metric, with entries g_{ij} = g(e_i, e_j), is exactly the coefficient array in the dual basis product expansion g = g_{ij} e^i ⊗ e^j, tying the abstract definition of a metric as a symmetric bilinear form directly to its concrete numerical representation in any chosen basis.