10.8.5 Tensor Vector Component Object Preservation

Tensor Vector Component Object Preservation is the property, guaranteed by the vector component change rule, that a vector reconstructed from its transformed components paired with the new basis vectors is identical to the vector reconstructed from its original components paired with the original basis vectors, so that the vector as an abstract object is left completely unaffected by any change of basis applied to its representation. It is the specific instance, restricted to rank-one contravariant tensors, of the more general tensor component law tensor preservation property, and it is what justifies treating a vector as a single fixed entity even though the numbers used to describe it change from basis to basis.

Statement of the Preservation Property

The Preserved Equality

Object preservation asserts that the two possible reconstructions of a vector, one using old components and the old basis, and the other using new components and the new basis, yield the same result.

Restriction to Vectors

While the same kind of preservation extends to tensors of any rank, this particular statement concerns specifically the rank-one contravariant case, in which a single upper index is transformed by the inverse change-of-basis matrix while the single corresponding basis vector transforms by the forward matrix.

Mechanism of Preservation

Cancellation of the Matrix and Its Inverse

Substituting the vector component change rule and the forward basis vector transformation into the new-basis reconstruction produces a product of the inverse matrix and the forward matrix, which reduces to the identity matrix upon summation over the shared index.

Once the identity matrix appears, only the original component paired with the original basis vector remains, confirming that the reconstruction has returned to its starting expression.

Dependence on the Contravariant Transformation Specifically

This cancellation only occurs because the vector's components are contracted with the inverse matrix while the basis vectors are contracted with the forward matrix; had the components instead used the forward matrix, matching the basis vectors' own transformation, the resulting reconstruction would not reduce to the identity and object preservation would fail.

Significance of the Property

Distinguishing Genuine Vectors From Arbitrary Arrays

An indexed array of numbers that fails to satisfy object preservation under a proposed transformation cannot represent a genuine vector, since its reconstructed value would depend on which basis happened to be chosen, contradicting the very notion of a vector as a single fixed object.

Supporting Basis-Independent Reasoning

Because object preservation holds for every valid change of basis, any statement made about a vector using its components in one basis can be translated into an equivalent statement using its components in any other basis, without altering the truth of the underlying claim about the vector itself.

Consistency Across Chains of Transformations

Because each individual change of basis preserves the vector, any sequence of successive changes of basis, each governed by its own instance of the vector component change rule, also preserves the vector overall, regardless of how many intermediate bases are involved.

Schematic Representation

The diagram shows both reconstructions, from old and new bases respectively, converging on the identical vector, the visual expression of object preservation under the vector component change rule.