4.16.1 Tensor Basis Rule Extension

Tensor Basis Rule Extension is the specific extension rule obtained from the general tensor multilinear extension construction in the particular case where the generating set used to specify the prescribed data is a genuine basis, so that no compatibility condition among the generators needs to be checked and the extension proceeds by a single, unconditional formula. It isolates the basis case as the simplest and most direct instance of extension construction, singling out the rule itself, the exact formula by which arbitrary arguments are mapped to outputs, rather than the broader question of when such a rule exists at all.

The Rule in Its Explicit Form

Stating the Rule Directly

Let V be an n-dimensional vector space with basis e_1, ..., e_n, and suppose scalars c(i_1, ..., i_p, j_1, ..., j_q) have been prescribed for every combination of basis indices. The basis rule extension is the assignment

with the coordinates α_r^{i_r} and v_s^{j_s} obtained from the unique basis expansions of the arguments. This is a single, closed-form rule, applicable directly to any collection of arguments without any preliminary verification step.

The Rule Requires No Consistency Check

Unlike the general extension construction applied to an arbitrary spanning set, the basis rule extension carries no compatibility condition to verify beforehand, since a basis admits no nontrivial linear relations among its elements; every possible prescription of scalars c(i_1, ..., j_q) is automatically compatible with the (empty) collection of relations among basis vectors, so the rule always succeeds in producing a well-defined tensor.

Deriving the Rule from the General Construction

Specializing the Spanning Set to a Basis

Applying the general extension construction with the spanning set g_1, ..., g_m specialized to g_r = e_r for r = 1, ..., n, and m = n, removes every linear relation that would otherwise need to be checked, since linear independence of the basis means the only relation ∑_r λ_r e_r = 0 is the trivial one with every λ_r = 0, which the compatibility condition satisfies vacuously regardless of the prescribed data.

The Rule as the Trivial-Compatibility Case

Because the compatibility condition of the general construction reduces to a vacuous statement whenever the generators form a basis, the basis rule extension can be regarded as the extension construction operating in its simplest possible regime, where the verification step is present in principle but never actually constrains the outcome.

Properties Inherited from the General Construction

Uniqueness of the Rule's Output

Because the basis expansion of any vector or covector is unique, the sum defining the basis rule extension involves exactly one term contributing the correct coordinate combination for each argument, and the resulting scalar is uniquely determined by the prescribed data c and the arguments supplied, matching the uniqueness guaranteed generally by tensor multilinear basis determination.

Multilinearity of the Rule

The basis rule extension, viewed as a function of its arguments, is multilinear in each slot for the same reason any instance of the general extension construction is multilinear once well-definedness is established: the rule is linear, to the first power, in the coordinates of each argument, guaranteeing both additivity and homogeneity throughout.

Using the Rule in Practice

As the Default Method for Defining a Tensor

Because it requires no consistency verification, the basis rule extension is the default and most commonly used method for defining a tensor from scratch: a practitioner need only choose a basis, prescribe the desired scalar for every combination of basis indices, and apply the rule directly to obtain a fully specified multilinear map.

Contrast with More General Extension Scenarios

When data is instead specified on an overcomplete spanning set rather than a basis, the basis rule extension does not apply directly; the general extension construction must be invoked instead, checking compatibility with every linear relation among the generators before a rule analogous to the basis rule extension can safely be applied.

Diagrammatic Summary

The diagram shows the basis rule extension applying directly from prescribed data on a basis to a fully defined tensor, bypassing the compatibility check required by the general extension construction whenever the generators are not independent.