5.9.5 Tensor Product Basis Expansion Role

Tensor Product Basis Expansion Role is the function that the induced basis of a tensor product plays in converting an arbitrary, abstractly given tensor into a unique numerical array of coefficients, and the corresponding role that array plays as the working representation on which nearly all concrete tensor computation is carried out.

The Expansion Statement

Given the induced basis of V1 ⊗ V2 ⊗ ⋯ ⊗ Vn, formed from chosen bases of the factor spaces, every element t of the tensor product expands uniquely as

with a single, uniquely determined scalar tk1…kn attached to every index tuple. This unique array is the coordinate representation of t relative to the chosen factor bases, and its assembly from the abstract element t is the basis expansion role in its most literal sense.

Uniqueness of the Expansion

That every tensor has exactly one such expansion, rather than several competing ones, is what makes the coordinate array a faithful, unambiguous stand-in for the abstract tensor.

Consequence of the Basis Property

Uniqueness of the expansion follows directly from the induced tensors forming a genuine basis: existence of an expansion is the spanning property, and uniqueness of the coefficients is the linear independence property, so the basis expansion role rests entirely on the two facts already established when the induced basis was constructed.

Coefficient Extraction Formula

When each factor space carries dual bases, the coefficient tk1…kn attached to a given index tuple can be recovered directly by applying the corresponding dual basis functionals to t, since these functionals were constructed (in establishing linear independence of the induced basis) to isolate exactly one basis element's coefficient at a time.

Expansion as an Isomorphism

Basis expansion is not merely a description of a tensor but an active, structure-preserving translation between the abstract tensor product space and the space of numerical arrays.

Linear Isomorphism to Coordinate Space

The map sending each tensor t to its coefficient array (tk1…kn) is a linear isomorphism from V1 ⊗ ⋯ ⊗ Vn onto the space of d1 × d2 × ⋯ × dn arrays over F, with linear structure (addition and scalar multiplication of tensors) corresponding exactly to entrywise addition and scaling of the coefficient arrays.

Role in Defining Tensor Operations Numerically

Because the expansion is an isomorphism, operations on abstract tensors — sums, scalar multiples, and, once suitably formulated, contractions and multilinear map evaluations — can be carried out entirely on the coefficient arrays without reference to the underlying abstract vector spaces, which is the entire basis for treating tensors computationally as multidimensional arrays of numbers.

Expansion of Decomposable Tensors

The basis expansion role takes a particularly clean form for decomposable tensors, whose coefficients follow the entrywise multiplicative pattern described elsewhere, but whose expansion still requires the same general apparatus to justify.

Expansion from Factor Coordinates

If a decomposable tensor v1 ⊗ ⋯ ⊗ vn has each factor vi expanded in its own basis with coefficients (vi)ki, distributing the tensor product over these expansions — using multilinearity — directly produces the basis expansion of the whole tensor, with coefficients tk1…kn equal to the product (v1)k1 ⋯ (vn)kn. This derivation is itself an application of the basis expansion role at the level of the individual factors, propagated up to the level of the full tensor product.

Expansion Under Change of Basis

Because the coefficients in the expansion depend on the chosen factor bases, the basis expansion role interacts directly with change-of-basis transformations.

Transformation of Coefficients

If the factor bases are changed via invertible matrices A1, …, An, the coefficient array of a fixed abstract tensor t transforms via the corresponding Kronecker product of the matrices (or their inverses, depending on convention), reflecting that the abstract tensor itself is unchanged while its numerical expansion is recomputed relative to the new bases.

Invariant Quantities Across Expansions

Certain quantities computed from a basis expansion, such as the tensor's norm under a compatible inner product, or its tensor rank, remain the same regardless of which basis was used to compute the expansion, distinguishing basis-dependent numerical coefficients from the basis-independent properties of the abstract tensor they represent.

Illustrative Diagram

The abstract tensor on the left expands, relative to the induced basis, into the numerical array on the right, with each cell holding one uniquely determined coefficient.