4.15.4 Tensor Multilinear Extension from Basis

Tensor Multilinear Extension from Basis is the constructive process of building the full multilinear map defined on all arguments, starting only from its prescribed values on basis vectors and basis covectors, by using additivity and homogeneity to extend those values step by step across every possible linear combination. It is the procedural counterpart to tensor multilinear basis determination, focusing on how the extension is actually carried out, argument by argument and slot by slot, rather than merely asserting that such an extension exists and is unique.

Extending One Slot at a Time

Starting from a Single-Slot Extension

Suppose the values of a candidate map are prescribed only on basis vectors in one particular slot, with every other slot already fixed to specific arguments. Extending to an arbitrary argument v = ∑_j v^j e_j in that slot proceeds by declaring

which is forced by additivity and homogeneity in that slot: additivity distributes the sum defining v across the map, and homogeneity pulls out each coordinate v^j as a scalar factor, leaving only the prescribed basis values T(..., e_j, ...) to be combined.

Repeating the Extension Across Every Slot

Once one slot has been extended to arbitrary arguments, the same procedure is applied to the next slot, still expressed in terms of basis elements, and then the next, until every one of the p + q slots has been extended in turn; after all slots have been processed, the resulting map accepts arbitrary arguments in every slot simultaneously, having been built up entirely from the originally prescribed basis values.

Order Independence of the Extension Procedure

Extending Slots in Any Sequence

The final extended map does not depend on the order in which the slots are processed during extension; because additivity and homogeneity in one slot do not interfere with the extension already carried out in another slot, extending the slots in the order 1, 2, ..., p+q produces exactly the same fully extended map as extending them in any other order.

Consistency with Direct Substitution into the Component Formula

The result of this slot-by-slot extension procedure coincides exactly with substituting the coordinates of every argument directly into the tensor multilinear component evaluation formula, using the prescribed basis values as the component array; the incremental, slot-by-slot construction and the all-at-once component substitution describe the same underlying extension, arrived at by two different routes.

Uniqueness of the Extension

No Other Multilinear Extension Is Possible

If a multilinear map T and a multilinear map T' agree on every basis input tuple, then the extension procedure applied to either one, starting from the same prescribed basis values, produces the identical fully extended map, since the procedure at each step is entirely forced by additivity and homogeneity, leaving no freedom for T and T' to diverge once their basis values coincide; this is exactly the uniqueness half of tensor multilinear basis determination, restated as a property of the extension procedure itself.

The Extension Cannot Be Chosen Freely Beyond the Basis Values

Once the basis values are fixed, every step of the extension procedure is determined without any further choice: additivity and homogeneity leave no room for an alternative rule that would still respect multilinearity, so the extension from a given set of basis values is the unique multilinear map compatible with those values.

Extension as a Realization of the Universal Property

Matching the Universal Property of the Tensor Product

The extension procedure realizes, in an elementary and computational form, the same universal property that characterizes the tensor product of vector spaces: any assignment of values on basis elements of the factors extends uniquely to a linear map out of the tensor product, and the slot-by-slot extension procedure is precisely how this unique linear map is constructed in coordinates.

Extension and the Reduced Arity Result

When only some slots have prescribed basis values and the rest remain entirely unconstrained placeholders, the extension procedure applied to just those prescribed slots produces a tensor reduced arity result, since the slots left unconstrained remain open exactly as in tensor multilinear partial evaluation, showing that extension from a basis and partial evaluation are complementary operations acting on the same underlying multilinear structure.

Practical Steps of the Extension

Building a Tensor from Scratch

In practice, extending a tensor from a basis begins by listing every basis input tuple, assigning each one its prescribed scalar value, and then applying the component evaluation formula whenever the tensor needs to be evaluated on new, non-basis arguments; this two-stage process, first fixing values on the basis and then extending as needed, is the standard method by which tensors are constructed and used in explicit computations.

Verifying an Extension Is Well-Formed

Before relying on an extension, it is standard to confirm that the prescribed basis values have been assigned consistently, meaning simply that every basis input tuple has received exactly one scalar value; because no further constraint is imposed on a basis value assignment, this check is the only verification needed before the extension procedure can proceed.

Diagrammatic Summary

The diagram shows the extension procedure advancing slot by slot, starting from prescribed basis values and ending with a fully extended multilinear map that accepts arbitrary arguments in every slot.