9.17.1 Tensor Basis Independent Object Identity

Tensor Basis Independent Object Identity is the principle that a tensor's identity as a distinct mathematical object is fixed by its abstract multilinear action alone, so that two descriptions arising from different bases represent the same tensor exactly when they are related by the correct transformation law, and represent different tensors otherwise. It addresses the question of when two component arrays, possibly expressed in different bases, should be regarded as descriptions of one and the same underlying tensor.

The Identity Criterion

Sameness Beyond Numerical Appearance

Two component arrays that look completely different numerically can nonetheless describe the same tensor, provided the arrays are related to each other by the transformation law corresponding to a valid change of basis between the two bases involved.

Sameness Requires the Full Transformation, Not Coincidence

Conversely, two component arrays that happen to be numerically identical while expressed relative to two genuinely different bases do not, in general, describe the same tensor, since matching numbers under mismatched bases does not satisfy the transformation law connecting them.

Establishing Identity in Practice

Direct Verification Through the Transformation Law

To confirm that two component arrays given in two different bases represent the same tensor, one applies the transformation law using the matrix relating the two bases and checks whether it converts one array exactly into the other.

Identity Through a Common Basis

Alternatively, identity can be established by transforming both arrays into a single common basis and checking that the resulting components agree exactly, since two tensors expressed identically in the same basis must be the same tensor.

Object Identity Versus Component Equality

Component Equality Is Basis Specific

Equality of components, checked directly number by number, is only a meaningful test of tensor identity when both arrays are already known to be expressed relative to the same basis. Comparing components from different bases directly, without first transforming to a common basis, is not a valid test.

Distinguishing Different Tensors of the Same Type

Two tensors of the same type, expressed relative to the same basis, are different tensors precisely when their component arrays differ in at least one entry, since equal components in a shared basis correspond to equal tensors by the uniqueness of the summation form.

Significance of Basis Independent Identity

A Stable Notion Underlying All Representations

Basis independent object identity provides the stable notion of what a tensor is that underlies every one of its many possible component representations, allowing statements like equality, addition, and comparison of tensors to be made sense of consistently regardless of which basis happens to be used at any given moment.

Preventing Misidentification Across Bases

Recognizing that object identity requires the transformation law, not mere numerical coincidence, guards against the error of mistaking two genuinely different tensors for the same one simply because their components happen to look alike when expressed in different, unrelated bases.