5.9.1 Tensor Product Basis Factor Selection

Tensor Product Basis Factor Selection is the process of choosing, for each vector space participating in a tensor product, an individual basis for that space, and then combining these choices systematically to produce a basis for the whole tensor product space out of the pairwise (or n-wise) combinations of the selected basis vectors.

Formal Description

Let V1, V2, …, Vn be finite-dimensional vector spaces over a field F, with respective dimensions d1, d2, …, dn. Basis factor selection means choosing, for every factor space Vi, a basis

The selected bases are then combined by tensoring: for every tuple of indices (k1, …, kn) with 1 ≤ ki ≤ di, the tensor e^(1)_k1 ⊗ e^(2)_k2 ⊗ ⋯ ⊗ e^(n)_kn is formed. The full collection of these tensors, over all possible index tuples, constitutes the induced basis of the tensor product V1 ⊗ V2 ⊗ ⋯ ⊗ Vn produced by the factor selection.

Why Selection Yields a Basis

The transition from n independent factor bases to a single basis of the tensor product relies on two properties that must both hold, and that the definition of the tensor product guarantees.

Spanning

Because every vector in each factor space Vi expands as a linear combination of the selected basis for Vi, and because the tensor product distributes over such linear combinations by multilinearity, every simple tensor — and hence, by the span role of simple tensors, every element of the whole tensor product — can be written as a linear combination of the selected combined basis tensors.

Linear Independence

The combined tensors e^(1)_k1 ⊗ ⋯ ⊗ e^(n)_kn are linearly independent precisely because the individual factor bases are linearly independent in their own spaces; this independence is a structural consequence of the tensor product construction (typically proved via the universal property, by constructing multilinear functionals that isolate one basis tensor at a time) rather than something that must be separately verified case by case.

Resulting Dimension

follows immediately, since the number of index tuples (k1, …, kn) is exactly the product of the individual dimensions.

Freedom and Constraint in the Selection

Not every aspect of basis factor selection is arbitrary: some choices are free, while the combination step imposes its own rigid structure once the individual choices are made.

Independence of Factor Choices

The basis chosen for each factor space Vi can be selected completely independently of the bases chosen for the other factor spaces; there is no compatibility condition linking, say, the basis of V1 to the basis of V2, since each factor contributes its own coordinate system to the tensor product without reference to the others.

No Freedom in the Combination Rule

Once the individual bases are fixed, the induced basis of the tensor product is entirely determined — it consists of exactly the tensors formed by taking one basis vector from each factor, with no additional choices available at the combination step. Any two people who select the identical bases for each Vi will therefore always produce the identical induced basis for the tensor product.

Effect of Reselection

Choosing a different basis for one or more of the factor spaces changes the induced tensor product basis according to the ordinary rules of change of basis, extended multilinearly across the factors.

Change of Basis Formula

If the basis of Vi is changed by an invertible matrix Ai, so that the new basis vectors are linear combinations of the old ones, the induced basis of the tensor product changes by the Kronecker product A1 ⊗_K A2 ⊗_K ⋯ ⊗_K An of the individual change-of-basis matrices, reflecting that each factor's reselection acts independently and combines multiplicatively at the level of coordinates.

Selection Adapted to Structure

In practice, bases are frequently selected not arbitrarily but to align with structure already present in each factor space — orthonormal bases when an inner product is available, eigenbases when a relevant linear operator acts on a factor, or bases adapted to a physical coordinate system — and the resulting induced tensor product basis inherits the corresponding structural properties (orthonormality, diagonalization of an induced operator, and so forth) automatically from the factorwise selection.

Selection in Computational Practice

Basis factor selection is the concrete mechanism by which abstract tensors are represented as numerical multidimensional arrays for storage and computation.

From Basis Selection to Array Indexing

Once bases are selected for every factor, any tensor in the product space is represented, relative to the induced basis, by a numerical array indexed by the tuple (k1, …, kn); this is precisely the multidimensional array representation used throughout numerical tensor computation, with the factor selection step supplying the meaning of each array index.

Consistency Across Computations

Because different basis selections induce different, related-but-distinct coordinate arrays for the same abstract tensor, computations that combine tensors from separate sources must ensure a consistent factor selection has been used throughout, or must explicitly track and apply the change-of-basis transformation before combining results.

Illustrative Diagram

Three basis vectors chosen for each of two factor spaces combine pairwise into nine basis tensors of the product space, illustrating how factor selection determines the induced basis by multiplying dimensions.