4.18.4 Tensor Multilinear Form Basis Dependence

Tensor Multilinear Form Basis Dependence is the phenomenon whereby the coordinate array representing a multilinear form changes when the bases of the underlying vector spaces change, even though the multilinear form itself, as an abstract map, remains exactly the same object. Basis dependence separates what is genuinely intrinsic to a multilinear form from what is an artifact of the particular coordinates chosen to describe it.

The Two Levels of Description

The Form Itself Is Basis-Independent

A multilinear form f: V₁ × ... × Vₙ → F is defined purely in terms of the vector spaces Vᵢ and the field F, with no reference to any basis. Two observers using different bases for the Vᵢ are describing the same map f; nothing about f as a function changes when a basis is swapped for another.

The Coordinate Array Is Basis-Dependent

Once bases {e^{(i)}_k} are chosen for each Vᵢ, f acquires a coordinate representation

as an array of scalars. Choosing a different basis for any Vᵢ produces a different array of numbers representing the same underlying form f; the array is a description relative to a chosen frame, not the form itself.

The Transformation Rule

Change of Basis on One Factor

Suppose the basis of Vᵢ is changed from {e^{(i)}_k} to a new basis {ê^{(i)}_k} related by ê^{(i)}_k = ∑_j P_{jk} e^{(i)}_j for a change-of-basis matrix P. Multilinearity of f in slot i forces the new coordinate array to relate to the old one by

with all other indices untouched. Because f acts linearly in slot i alone, the transformation involves only the change-of-basis data for Vᵢ, contracted against the index corresponding to that slot, leaving indices belonging to the other slots unaffected.

Simultaneous Change on All Factors

If every factor space changes basis simultaneously, the array transforms by contracting each index against the corresponding change-of-basis matrix for its own factor, one contraction per index, applied independently. This index-by-index transformation law is precisely the classical description of a "tensor" as a quantity that "transforms according to certain rules" under a change of coordinates, with the multilinear form providing the coordinate-free object those rules are describing.

What Is Invariant Despite Basis Dependence

Values on Actual Vectors

Although the array T changes with the basis, the value f(v₁, ..., vₙ) for any specific vectors v₁, ..., vₙ is unchanged, since the change of coordinates of the vectors themselves and the change of coordinates of the array cancel exactly, by construction, to leave the scalar output the same. Basis dependence affects only the intermediate bookkeeping, never the final evaluated result.

Symmetry and Antisymmetry Type

Whether a form is symmetric, alternating, or neither is a property preserved under any change of basis, since permuting arguments and permuting the corresponding array indices commute with any fixed change-of-basis transformation applied to each factor independently. A form symmetric in one basis's coordinates remains symmetric in every other basis's coordinates.

Rank of a Bilinear or Multilinear Form

For a bilinear form, the rank of the associated matrix, the dimension of its image as a linear map (after identifying one factor with its dual), is invariant under change of basis on either factor, even though the individual matrix entries change; more generally, for a multilinear form viewed as an element of a dual tensor product, its tensor rank, the minimal number of elementary tensors needed to express it, is a basis-independent invariant despite the individual coordinates changing.

Contrast With the Universal Property Definition

The Universal Property Is Stated Without Bases

The universal property characterizing the tensor product, and hence characterizing multilinear forms as elements of a dual tensor product space, makes no reference to any basis: it is phrased entirely in terms of factorization of multilinear maps through the canonical map ⊗. This is deliberate, since it guarantees that any conclusion drawn from the universal property applies equally regardless of which basis, if any, is later chosen for computation.

Coordinates as a Computational Convenience

Basis-dependent coordinate arrays are introduced only for computation: they allow a multilinear form on finite-dimensional spaces to be specified and manipulated using finitely many numbers, and allow standard linear-algebraic algorithms, matrix multiplication, determinant computation, Gaussian elimination, to be applied to problems about multilinear forms. The basis-free definition remains the authoritative one; the coordinate array is a chosen representative of the form relative to a frame, useful for calculation but not part of the form's intrinsic content.

Practical Implications

Verifying Basis-Independence of a Construction

When defining a new operation on multilinear forms via a formula involving indices, such as a contraction or a product, verifying that the construction is well defined typically requires checking that the result transforms correctly under change of basis, that is, that applying the construction commutes with the transformation rule for each factor; a construction that fails this check is not a legitimate basis-independent operation, however plausible its coordinate formula may look.

Choosing Convenient Bases

Because the underlying form does not depend on the basis, a basis can always be chosen for computational convenience, such as an orthonormal basis diagonalizing a symmetric bilinear form, without altering any conclusion about the form itself; basis dependence of the coordinate array is precisely what licenses this freedom, since any basis gives a faithful, if differently expressed, description of the same underlying multilinear object.