12.7.4 Tensor Equality Basis Independent Criterion

Tensor Equality Basis Independent Criterion is the standard by which two tensors are judged equal based on their identity as abstract multilinear objects rather than on the coincidence of their numerical components in any single, arbitrarily chosen basis, so that equality established in one basis is guaranteed to hold in every other basis as well.

Formulating the Criterion

Equality as a Property of the Object, Not the Description

Since a tensor exists independently of any particular basis, and its component array in a given basis is only one possible description of it, the basis independent criterion states that two tensors $$A$$ and $$B$$ are equal exactly when they are the same abstract object, which can be verified by checking componentwise agreement in any one basis, not necessarily every basis individually.

Sufficiency of a Single Basis Check

If the components of $$A$$ and $$B$$ agree in one chosen basis:

for every index combination, the criterion guarantees this same equality holds after transforming to any other basis, without needing to independently verify it there.

Proof of Basis Independence

Linearity of the Transformation Law

Both $$A$$ and $$B$$, sharing the same type, transform under a change of basis according to the identical linear rule $$T$$. If their components agree before transformation, applying the same linear map to both sides preserves that agreement:

Since $$T$$ is a function, applying it to two inputs already known to be equal necessarily yields equal outputs, which is exactly the reasoning underlying the basis independent criterion.

Reduction to Zero Tensor Argument

Equivalently, the criterion can be justified by observing that if $$A-B=0$$ in one basis, then since the zero tensor transforms to the zero tensor under any change of basis, $$A-B$$ remains the zero tensor in every basis, which is equivalent to $$A=B$$ holding universally.

Practical Significance

Freedom to Choose a Convenient Basis

Because of this criterion, verifying tensor equality can always be carried out in whichever basis makes the computation simplest, such as an orthonormal basis or a basis aligned with some natural symmetry of the problem, without any loss of generality.

Distinguishing Genuine Equality from Coordinate Artifacts

The criterion also protects against mistakenly treating a coordinate-dependent numerical coincidence as meaningful equality, since a basis independent check ensures the agreement reflects the tensors themselves rather than a special property of one particular coordinate system.

Contrast with a Naive Componentwise Check

Naive Check Without Considering Basis

Comparing raw numbers from two tensors that happen to be expressed in different bases, without first transforming to a shared basis, does not constitute a valid application of the criterion and can produce incorrect conclusions about equality.

Correct Application of the Criterion

The basis independent criterion is correctly applied only once both tensors are expressed relative to the same basis, at which point componentwise agreement becomes a valid and sufficient test that will remain valid under any subsequent change of basis.

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