12.18.4 Tensor Operation Basis Compatibility

Tensor Operation Basis Compatibility ensures algebraic consistency across different bases, enabling accurate transformations and computations in tensor algebra.

Tensor Operation Basis Compatibility is the requirement that, whenever a tensor operation is carried out using explicit coordinates, the bases chosen for every tensor and every auxiliary map involved be consistently related — either literally identical where the operation demands shared indices, or connected by a specified, known change-of-basis transformation where the operation's own construction supplies that connection. It addresses the layer of compatibility that only becomes visible once abstract tensor operations are reduced to concrete coordinate computations, complementing variance type compatibility and ambient space compatibility, which can both be satisfied while a basis-level mismatch still renders a coordinate computation meaningless.

Shared Basis Requirement for Direct Coordinate Operations

Contraction Requires a Common Index Range

\sum_{k} T_{\dots k \dots}^{\dots k \dots}

Contracting a contravariant index against a covariant index by summing over a shared label k presupposes that both indices are expressed relative to dual bases — a basis {eᵢ} of V and its dual basis {εʲ} of V*, matched so that εʲ(eᵢ) = δᵢʲ. Using a covariant index expressed relative to some other, unrelated basis of V* not dual to {eᵢ} invalidates the usual contraction formula, even though the variance types and ambient space are otherwise perfectly compatible.

Addition and Comparison Require Identical Bases

T + S = \sum_{i} (T^{i} + S^{i}) e_{i}

Adding two tensors coordinate-by-coordinate is only correct when both tensors' coordinates Tⁱ and Sⁱ are expressed relative to the identical basis {eᵢ}; adding coordinates taken from two different bases without first converting one into the other's basis produces numbers that do not correspond to any single well-defined tensor.

Explicit Change-of-Basis Requirement for Transport Operations

Pullback and Pushforward Carry Their Own Basis-Connecting Data

f (e_{j}) = \sum_{i} M_{i j} g_{i}

The pullback and pushforward operations do not require the bases of their source and target spaces to be identical; instead, they require the matrix M representing the source map f relative to the chosen bases {eⱼ} of V and {gᵢ} of W to be explicitly known, since this matrix is exactly what the coordinate form of the slot transformation and vector slot action apply. Basis compatibility here means the relationship between the two bases is specified through f, not that the bases themselves coincide.

Extension of Scalars Requires the Source Basis to Transport Cleanly

When forming V ⊗_R S, the basis {eᵢ} of the source domain transports to the basis {eᵢ ⊗ 1} of the target domain only if {eᵢ} is genuinely a basis over R in the first place; using a spanning set that is not linearly independent over R as though it were a basis produces a placement that does not correctly describe the extended domain's actual coordinate structure.

Diagram of Basis Compatibility Requirements

Consequences of Basis Incompatibility

Silently Wrong Numerical Results

Unlike variance type or ambient space incompatibility, which typically produce an expression with no valid formula to evaluate at all, basis incompatibility often permits a computation to proceed and produce numbers — numbers that simply do not correspond to any coherent tensor, because they mix coordinates expressed relative to unrelated bases. This makes basis incompatibility a more insidious failure mode, since it does not announce itself as clearly as a type or space mismatch.

Detection Requires Explicit Basis Bookkeeping

Because two different bases can assign identical-looking coordinate labels (such as both being indexed 1, 2, 3) to genuinely different underlying vectors, verifying basis compatibility requires tracking which specific basis each set of coordinates is relative to, not merely confirming that index ranges match numerically.

Establishing Basis Compatibility Where None Initially Exists

Explicit Change-of-Basis Conversion

When two tensors are given in different, unrelated bases of the same space, basis compatibility for addition or contraction can be established by explicitly converting one tensor's coordinates into the other's basis using the appropriate change-of-basis matrix, after which the ordinary coordinate formulas for addition or contraction apply validly.

Choosing Mutually Adapted Bases in Advance

For operations such as contraction that require dual bases, or transport operations such as pullback and pushforward that require an explicit basis-relating matrix, basis compatibility is most efficiently established by choosing the bases involved together, from the outset, so that the required duality or the required matrix representation of the connecting map is simple or already known, rather than attempting to reconcile independently chosen bases after the fact.