5.23.3 Tensor Product Basis Component Expansion

Tensor Product Basis Component Expansion is the procedure of taking an arbitrary element of V ⊗ W, however it happens to be presented (typically as some sum of simple tensors that are not yet aligned with any chosen basis), and reducing it to its unique expansion Σ c_{ij}(eᵢ ⊗ fⱼ) in terms of a fixed basis, together with the proof that this reduction always exists and is unique. This expansion procedure is what makes the abstract fact "simple tensors eᵢ ⊗ fⱼ form a basis of V ⊗ W" into a usable computational tool for converting any given tensor into standard coordinate form.

Existence of the Expansion

Reducing an Arbitrary Sum of Simple Tensors

An arbitrary element t ∈ V ⊗ W is, by construction, some finite sum t = Σₖ vₖ ⊗ wₖ. Expanding each vₖ = Σᵢ aₖᵢ eᵢ and each wₖ = Σⱼ bₖⱼ fⱼ in the chosen bases and distributing ⊗ bilinearly over both sums produces

so t is expressed in the basis {eᵢ ⊗ fⱼ} with components c_{ij} = Σₖ aₖᵢ bₖⱼ, regardless of how many simple tensors were in the original presentation of t.

Existence Guaranteed Regardless of Original Presentation

Because this reduction works for any finite sum of simple tensors, the expansion in the basis {eᵢ ⊗ fⱼ} exists no matter how t was originally given — whether as a single simple tensor, an arbitrary finite sum, or the output of some other tensor construction — confirming existence of the basis expansion in full generality.

Uniqueness of the Expansion

Linear Independence Forces Uniqueness

Since {eᵢ ⊗ fⱼ} is a basis (not merely a spanning set), the coefficients c_{ij} obtained above are the only possible coefficients expressing t in this basis: if t = Σ c_{ij}(eᵢ ⊗ fⱼ) = Σ c′_{ij}(eᵢ ⊗ fⱼ), then Σ (c_{ij} − c′_{ij})(eᵢ ⊗ fⱼ) = 0, and linear independence of the basis forces every c_{ij} − c′_{ij} = 0.

Consequence: Well-Defined Coordinates

Uniqueness means the expansion procedure defines a genuine function from V ⊗ W to the space of m × n coefficient arrays, associating to every tensor exactly one array of coordinates relative to the chosen basis — a prerequisite for coordinates to be meaningful at all.

Diagram of the Expansion Procedure

A Worked Example

Setting Up the Example

Let V = W = ℝ² with standard basis {e₁, e₂}. Consider the tensor presented as the sum t = (e₁ + 2e₂) ⊗ e₁ + e₂ ⊗ (3e₁ − e₂), not yet in the canonical basis form.

Carrying Out the Expansion

Distributing bilinearly, the first term contributes e₁ ⊗ e₁ + 2(e₂ ⊗ e₁), and the second term contributes 3(e₂ ⊗ e₁) − e₂ ⊗ e₂. Collecting coefficients of each basis element eᵢ ⊗ eⱼ gives

so the expanded component matrix is [[1, 0], [5, −1]], obtained purely by collecting like terms after distributing the original presentation over the basis.

Expansion Under a Different Basis Choice

The Same Tensor, Different Coordinates

If a different basis is chosen for V or W, the same tensor t receives a different, but equally valid and equally unique, component array in that new basis; the expansion procedure applies identically, only the input bases change, and the change-of-basis transformation rule relates the two resulting arrays.

Expansion Is Basis-Relative, Not Basis-Free

This dependence on the chosen basis is expected and does not conflict with uniqueness: uniqueness holds relative to a fixed basis, while the numerical values of the components themselves are basis-relative, a distinction fundamental to correctly interpreting any component expansion.

Significance of Basis Component Expansion

Turning Abstract Tensors into Computable Data

Basis component expansion is the procedure that converts a tensor given in any form whatsoever into the single, standard, and unique coordinate representation needed for numerical computation, comparison, and storage, making it the practical endpoint of the abstract theory of tensor product bases.

Prerequisite for All Coordinate-Based Tensor Operations

Every operation that manipulates tensors through their components — applying a tensor product of linear maps, computing rank, checking symmetry, performing contractions — implicitly relies on this expansion procedure having already reduced the tensors involved to their unique basis coordinates.