5.16.2 Tensor Product Formal Linear Combination

The tensor product formal linear combination constructs multilinear relationships by combining vectors from different spaces into a new structured algebraic object.

Tensor Product Formal Linear Combination is a scalar-weighted finite sum of simple tensors, Σ cᵢ (vᵢ ⊗ wᵢ), considered as the standard vector-space-level expression for an element of V ⊗ W, distinguished from the plain formal sum Σ vᵢ ⊗ wᵢ by making the coefficient scalars explicit rather than absorbing them into one of the two factors. This distinction matters because the tensor product is, after all, a vector space, and its elements are properly expressed as linear combinations of generators with scalar coefficients, exactly as in any other vector space, even though the generators themselves already have internal bilinear structure.

From Formal Sum to Formal Linear Combination

Absorbing Coefficients into a Factor

Because of the homogeneity relation, a coefficient c multiplying a simple tensor can always be absorbed into either factor:

c (v \otimes w) = (c v) \otimes w = v \otimes (c w)

This is precisely why a plain formal sum Σ vᵢ ⊗ wᵢ, without explicit coefficients, still represents every element of V ⊗ W: any needed scalar has already been folded into one of the vectors.

Why the Explicit Coefficient Form Is Still Useful

Despite this redundancy, writing a tensor as an explicit linear combination Σ cᵢ (vᵢ ⊗ wᵢ) with vᵢ, wᵢ drawn from fixed bases, and coefficients cᵢ collected separately, is the standard form used when treating V ⊗ W purely as an abstract vector space, matching the way any vector space element is written as a linear combination of basis vectors.

The Canonical Linear Combination Relative to Bases

Expansion in Terms of Basis Tensors

Given bases {eᵢ} of V and {fⱼ} of W, every element of V ⊗ W has a unique expression as the formal linear combination:

t = \sum_{i, j} c_{i j} (e_{i} \otimes f_{j})

with scalar coefficients cᵢⱼ ∈ F uniquely determined by t, since {eᵢ ⊗ fⱼ} forms a basis of V ⊗ W.

Comparison with an Arbitrary Formal Linear Combination

An arbitrary formal linear combination, not necessarily using basis vectors, Σ cᵢ (vᵢ ⊗ wᵢ), can always be rewritten in the canonical basis form by expanding each vᵢ and wᵢ in the chosen bases and collecting terms, illustrating that the canonical form is a special, coefficient-unique case of the general formal linear combination.

Linear Combination Rules Applied to Tensors

Addition of Formal Linear Combinations

Two formal linear combinations are added by combining like basis-tensor terms, exactly as with ordinary vectors:

\sum_{i, j} a_{i j} (e_{i} \otimes f_{j}) + \sum_{i, j} b_{i j} (e_{i} \otimes f_{j}) = \sum_{i, j} (a_{i j} + b_{i j}) (e_{i} \otimes f_{j})

Scalar Multiplication of Formal Linear Combinations

Multiplying by a scalar k simply scales every coefficient in the combination:

k \sum_{i, j} c_{i j} (e_{i} \otimes f_{j}) = \sum_{i, j} (k c_{i j}) (e_{i} \otimes f_{j})

Diagram of the Coefficient Array

Significance of the Linear Combination Viewpoint

Bridge to Standard Linear Algebra

Treating tensors as formal linear combinations with coefficients arranged in an array is exactly what identifies V ⊗ W with the space of matrices Mat(dim V, dim W; F) once bases are fixed, connecting the abstract tensor product directly to familiar matrix computations, including rank, trace, and eigenvalue analysis when V = W.

Foundation for Computational Tensor Algebra

In numerical and symbolic computation, tensors are almost always stored and manipulated as coefficient arrays relative to some chosen basis, making the formal linear combination viewpoint, rather than the abstract quotient construction, the practical starting point for algorithms operating on tensor product spaces.