14.17.5 Tensor Map Product Formula Basis Dependence

Understanding how tensor map products depend on basis choices in algebraic structures.

Tensor Map Product Formula Basis Dependence is the property that the explicit numerical formula computing the components of a tensor product map $f \otimes g$ changes whenever the bases chosen for the domain and codomain spaces change, even though the linear map $f \otimes g$ itself, as an abstract object, does not depend on any choice of basis at all. It separates two things that are easy to conflate: the map, which is basis-free, and the array of numbers used to represent that map once bases have been fixed, which is basis-bound and transforms according to a definite rule when the bases are replaced.

Setup and Notation

Vector Spaces and Linear Maps

Let $V$ , $V^{'}$ , $W$ , and $W^{'}$ be finite-dimensional vector spaces over a field $K$ , and let

f : V \to V^{'}

g : W \to W^{'}

be linear maps. Their tensor product is the unique linear map

f \otimes g : V \otimes W \to V^{'} \otimes W^{'}

satisfying $(f \otimes g) (v \otimes w) = f (v) \otimes g (w)$ for all $v \in V$ and $w \in W$ . This map is defined by its action on simple tensors, without reference to any basis, which is exactly why the map itself is a basis-independent object.

Component Formula Relative to a Basis

Once bases ${e_{i}}$ of $V$ , ${e_{a}'}$ of $V^{'}$ , ${f_{j}}$ of $W$ , and ${f_{b}'}$ of $W^{'}$ are fixed, the maps $f$ and $g$ acquire component matrices $F_{i}^{a}$ and $G_{j}^{b}$ , and the tensor product map acquires the component formula

{(f \otimes g)}^{a b}__{i j} = F_{i}^{a} G_{j}^{b}

so that the coefficient attached to the basis tensor $e_{i} \otimes f_{j}$ mapping to $e_{a}' \otimes f_{b}'$ is simply the product of the two individual component entries. This is the object whose dependence on basis choice is the subject of this page.

Basis Dependence of the Component Array

Change of Basis on the Factors

Suppose the bases of $V$ and $W$ are replaced according to invertible change of basis matrices $P$ and $Q$ , so that new basis vectors are given by

{\tilde{e}}_{k} = \sum_{i} P_{k}^{i} e_{i}

{\tilde{f}}_{l} = \sum_{j} Q_{l}^{j} f_{j}

and, correspondingly, the matrices of $f$ and $g$ transform by conjugation with $P$ and $Q$ and the analogous matrices for the codomain bases.

Transformation Law for the Tensor Product Components

Because the component formula for $f \otimes g$ is a plain product of the components of $f$ and the components of $g$ , its transformation under a simultaneous change of basis is the product of the two separate transformation laws:

{\tilde{F}}_{} = P F P^{- 1}

\tilde{G} = Q G Q^{- 1}

\tilde{F} \otimes \tilde{G} = (P \otimes Q) (F \otimes G) {(P \otimes Q)}^{- 1}

The array of numbers representing $f \otimes g$ is therefore conjugated by the Kronecker product $P \otimes Q$ of the individual change-of-basis matrices, not by an independently chosen matrix. This is the precise sense in which the formula is basis dependent: every entry of the array is tied to the specific bases used, and replacing the bases forces a specific, computable replacement of every entry.

Kronecker Product Representation

When bases are ordered lexicographically, the matrix of $f \otimes g$ is exactly the Kronecker product of the matrices of $f$ and $g$ ,

[f \otimes g] = [f] \otimes [g]

and the well-known identity $(A \otimes B) (C \otimes D) = (A C) \otimes (B D)$ is precisely what reduces the change-of-basis conjugation above to the product of the two individual conjugations. Different orderings of the combined basis of $V \otimes W$ permute the rows and columns of this Kronecker product, which is itself a further, purely combinatorial, form of basis dependence distinct from the change-of-basis dependence described above.

Invariance of the Underlying Map

Coordinate-Free Definition Does Not Reference Components

The map $f \otimes g$ is defined by its action on tensors, $v \otimes w \mapsto f (v) \otimes g (w)$ , and this rule makes no reference to coordinates. Consequently the map itself, as a function between the abstract vector spaces, is the same map regardless of which bases are later chosen to write it down. Only the array of numbers used to record that map in a particular coordinate system depends on the basis.

Consistency Under Basis Change

For any vector $v \otimes w$ , computing its image through the original components and through the transformed components yields the same vector expressed in the two different coordinate systems; the transformation law of the previous section is exactly the condition needed to guarantee this agreement. This is the formal statement that basis dependence of the formula and basis independence of the map are compatible: the formula changes precisely so that the map it computes does not.

Explicit Example

Two-Dimensional Case

Let $V$ , $V^{'}$ , $W$ , and $W^{'}$ all be two-dimensional, with $f$ and $g$ both represented in the original bases by the matrix

[\begin{matrix} 1 & 0 \\ 0 & 2 \end{matrix}]

so that the four-by-four component array of $f \otimes g$ is the Kronecker product of this matrix with itself, a diagonal matrix with entries $1$ , $2$ , $2$ , $4$ . If the basis of $V$ is changed by an invertible matrix $P$ that is not diagonal, the new component array of $f$ generally acquires off-diagonal entries, and correspondingly the new sixteen-entry array of $f \otimes g$ acquires off-diagonal entries determined exactly by $(P \otimes I) (F \otimes G) {(P \otimes I)}^{- 1}$ , illustrating concretely how the same underlying map is recorded by two entirely different numerical arrays depending on the basis in force.