4.15.5 Tensor Map Determination by Basis Values

Tensor maps are uniquely determined by their action on basis elements, a fundamental principle in tensor algebra that underpins coordinate-free computations.

Tensor Map Determination by Basis Values is the theorem asserting that two multilinear maps of the same type, defined on the same vector space, must be identical as functions on all arguments whenever they agree on every combination of basis vectors and basis covectors. It is the precise uniqueness statement underlying tensor multilinear basis determination, formulated here as a standalone theorem with an explicit proof, isolating the logical argument that guarantees a tensor cannot be altered without altering at least one of its basis values.

Statement of the Theorem

The Uniqueness Claim

Let V be a finite-dimensional vector space with basis e_1, ..., e_n, and let T and T' be two type (p, q) tensors on V. If

T (e^{i_{1}}, \dots, e_{j_{q}}) = T^{'} (e^{i_{1}}, \dots, e_{j_{q}})

for every combination of indices i_1, ..., i_p, j_1, ..., j_q, each ranging from 1 to n, then T(v_1, ..., v_{p+q}) = T'(v_1, ..., v_{p+q}) for every choice of arguments v_1, ..., v_{p+q} drawn from V and V* as required by the type of T.

Reformulation as a Difference Map

Equivalently, defining D = T - T' as the difference of the two tensors, a genuine tensor by ordinary vector space addition and scalar multiplication on the space of multilinear maps, the theorem states that if D vanishes on every basis input tuple, then D vanishes identically on every input tuple whatsoever.

Proof of the Theorem

Reduction to the Difference Map

It suffices to prove the reformulated statement about D, since T = T' on all arguments is equivalent to D being the zero map on all arguments. By hypothesis, D vanishes on every basis input tuple.

Expanding an Arbitrary Argument

Take an arbitrary input tuple v_1, ..., v_{p+q}, and expand each argument v_l in the basis, v_l = ∑_{k} v_l^{k} e_k (with e^k in place of e_k for contravariant slots). Applying the tensor multilinear additivity property and the tensor multilinear homogeneity property of D, one slot at a time, distributes D across every term of every expansion:

D (v_{1}, \dots, v_{p + q}) = \sum_{k_{1}, \dots}^{⋮} v_{1}^{k_{1}} \dots D (e_{k_{1}}, \dots)

Concluding the Proof

Every term on the right-hand side involves D evaluated on a basis input tuple, and by hypothesis every such term is zero. Since the sum has finitely many terms, all of them zero, the entire sum is zero, so D(v_1, ..., v_{p+q}) = 0. As the input tuple was arbitrary, D vanishes on every input tuple, completing the proof.

Why Each Hypothesis Is Necessary

The Necessity of Finite Dimensionality

The proof relies on expanding each argument as a finite sum over a basis of V; this expansion is only guaranteed to exist, with finitely many terms, because V is finite-dimensional, which is why the theorem as stated requires V to be finite-dimensional.

The Necessity of Multilinearity, Not Merely Equality on Basis Elements

The theorem would fail for arbitrary functions of several variables that merely happen to agree on basis elements; it is specifically the multilinearity of D, invoked through additivity and homogeneity at every step of the expansion, that allows the vanishing on basis input tuples to propagate to vanishing everywhere.

Consequences of the Theorem

A Complete Classification Tool

Because two tensors are equal precisely when their basis values coincide, the theorem reduces the question of whether two tensors are the same object to a finite, checkable comparison of n^{p+q} numbers, rather than requiring a comparison across the entire, generally infinite, domain of possible arguments.

Foundation for Basis-Dependent Component Bookkeeping

The theorem justifies treating a tensor's component array as a faithful and complete stand-in for the tensor itself in any fixed basis, since no distinct tensor can share the same component array, which is the logical basis for all subsequent component-based manipulation of tensors, including addition, scalar multiplication, and contraction performed directly on components.

Diagrammatic Summary

The diagram traces the logical chain of the proof, from agreement on basis values, to the vanishing of the difference tensor on basis input tuples, to the vanishing of that difference everywhere, establishing that the two original tensors are identical.