4.15 Tensor Multilinear Basis Determination

Tensor Multilinear Basis Determination is the principle that a multilinear map is completely and uniquely determined by its values on every combination of basis vectors and basis covectors drawn from a fixed basis of the underlying vector space, so that no additional information beyond these basis values is needed to recover the map's behavior on arbitrary arguments. It is the theoretical foundation underlying the component description of a tensor, explaining precisely why listing a finite table of numbers, one for each combination of basis indices, suffices to specify a multilinear map completely.

Statement of the Determination Principle

Basis Values Suffice

Let V be a finite-dimensional vector space with basis e_1, ..., e_n, and let e^1, ..., e^n be the dual basis of V*. For a type (p, q) tensor T, the values

taken over every combination of indices ranging from 1 to n, completely determine T on every possible choice of arguments, not merely on basis vectors and basis covectors themselves.

Uniqueness of the Recovered Map

If two type (p, q) tensors T and T' agree on every combination of basis vectors and basis covectors, meaning all of their basis values coincide, then T and T' are equal as multilinear maps, agreeing on every possible argument, not only on basis elements; basis determination guarantees there is no way for two distinct multilinear maps to share every one of their basis values.

Why Additivity and Homogeneity Force This Determination

Expanding Arbitrary Arguments in the Basis

Any argument supplied to T, whether a vector or a covector, can be written as a linear combination of basis elements. Repeated application of the tensor multilinear additivity property and the tensor multilinear homogeneity property, one slot at a time, distributes T across every term of every such expansion, converting an evaluation on arbitrary arguments into a sum of evaluations on combinations of basis elements, weighted by the coordinates of the arguments.

The Component Evaluation Formula as the Mechanism

This distribution process is exactly what produces the tensor multilinear component evaluation formula: since additivity and homogeneity guarantee this distribution is always valid, and since the resulting sum involves only the basis values of T multiplied by known coordinates, the value of T on any input tuple is fully computable from its basis values alone, with no further information about T required.

Existence Alongside Uniqueness

Any Choice of Basis Values Defines a Tensor

Basis determination is not only a uniqueness statement but also an existence statement: given an arbitrary assignment of a scalar to every combination of basis indices, there exists a unique multilinear map whose basis values match that assignment exactly, obtained by declaring the component evaluation formula, built from the chosen assignment, to be the definition of the map on all arguments.

Well-Definedness of the Constructed Map

Verifying that this constructed map is genuinely multilinear amounts to checking that the component evaluation formula, viewed as a function of the coordinates of the arguments, satisfies additivity and homogeneity in each slot; because the formula is linear in each set of coordinates by construction, these checks succeed automatically, confirming that any assignment of basis values does define a legitimate tensor.

Consequences of Basis Determination

Finite Specification of an Infinite-Domain Map

Although a multilinear map is defined on the entire, generally infinite, set of possible input tuples, basis determination shows that specifying it requires only the finite table of n^{p+q} basis values, since every other value follows automatically from these through the component evaluation formula; this finite reducibility is what makes tensors practical to store, manipulate, and compute with.

Change of Basis and Redetermination

If a different basis is chosen for V, the same tensor T is determined by a different table of basis values, related to the original table by the standard tensor transformation law; basis determination holds relative to any basis chosen, and the specific numbers making up the determining table change accordingly, while the abstract tensor T itself remains unchanged throughout.

Basis Determination and Partial Evaluation

Determining Reduced Arity Results

The tensor reduced arity result obtained by partial evaluation is likewise fully determined by its own basis values, restricted to the open slots, since it is itself a multilinear map on those slots and the same determination principle applies to it directly, using the same additivity and homogeneity properties inherited from the original tensor T.

Building Up Determination Incrementally

Basis determination can be established incrementally, slot by slot, by fixing each argument to a basis element in turn and applying tensor multilinear slot substitution; after all slots have been fixed to basis elements in every possible combination, the resulting collection of scalars is exactly the basis value table that determines the tensor.

Diagrammatic Summary

The diagram shows the finite table of basis values on the left determining, through additivity and homogeneity, the entire multilinear map on the right, valid for every possible choice of arguments.