2.19.2 Tensor Basis Independent Vector Space

Tensor basis independent vector spaces abstract linear algebra by focusing on structure, not specific bases, enabling broader mathematical and physical applications.

Tensor Basis Independent Vector Space is a vector space considered together with the guarantee that any tensor, tensor operation, or tensor-derived quantity built on it retains a well-defined meaning that does not depend on which basis was used to compute it, even though a basis may be freely introduced and used for numerical work. Where a coordinate-free treatment defines vectors and operations without ever mentioning coordinates, basis independence addresses the complementary and equally essential question of what happens once coordinates are introduced: it identifies precisely which numerical expressions, built from components in a chosen basis, secretly describe the same invariant object regardless of that choice, and it supplies the transformation law that certifies this invariance.

The Central Problem Basis Independence Solves

Components Look Different, the Object Does Not

Once a basis e_1, ..., e_n of a vector space V is fixed, a vector v acquires components v^1, ..., v^n, and switching to a different basis e'_1, ..., e'_n produces a different list of numbers v'^1, ..., v'^n for the very same vector v. Basis independence is the property that, despite this numerical change, there is one and only one underlying vector being described, and a definite rule connects the two component lists.

The Transformation Law as the Certificate of Invariance

For a type (p, q) tensor, basis independence is guaranteed precisely because its components transform according to the standard rule under a change of basis matrix A:

{\tilde{T}}_{l_{1} \dots l_{q}}^{k^{1} \dots k^{p}} = A_{i_{1}}^{k^{1}} \dots {(A^{- 1})}_{l_{1}}^{j^{1}} \dots T_{j_{1} \dots}^{i^{1} \dots}

Any array of numbers attached to a basis that obeys this exact rule under every possible change of basis is, by definition, describing one basis-independent tensor; an array that fails to obey it is not a tensor at all, merely a basis-dependent list of numbers with no invariant meaning.

Distinguishing Invariant Quantities From Basis-Dependent Ones

Scalars Extracted From Tensors

A quantity derived from a tensor is basis independent exactly when it is built using operations, such as contraction, that are compatible with the transformation law and end with no free indices remaining. The trace of a type (1,1) tensor, obtained by contracting its single upper index against its single lower index, is such a quantity: it produces the same number regardless of the basis used to compute the individual diagonal entries being summed.

tr (T) = T_{i}^{i}

with the repeated index summed, and this sum is provably identical whether computed from T or from T~.

Individual Components Are Not Invariant

By contrast, a single component T^1_1 of a tensor, taken in isolation, has no basis-independent meaning: it generally takes a different numerical value in a different basis, and no rule of tensor algebra treats an isolated component as a scalar. Basis independence is thus a property of whole tensors and of properly contracted combinations of tensors, never of individual entries picked out of a component array.

Canonical Examples of Basis-Independent Objects

The Identity Endomorphism and the Kronecker Delta

The identity linear map on V, viewed as a type (1,1) tensor, has components equal to the Kronecker delta δ^i_j in every basis simultaneously, since the transformation law applied to δ returns δ unchanged for any invertible A. This makes the identity tensor the simplest nontrivial example of a basis-independent object whose component description happens not to vary at all across bases.

Eigenvalues as Basis-Independent Invariants of a (1,1) Tensor

The eigenvalues of a type (1,1) tensor, viewed as a linear operator, do not depend on the basis used to write down its matrix, even though the matrix entries themselves change completely under a change of basis; this is because eigenvalues are defined through the characteristic polynomial, a quantity built by contraction and determinant operations that are themselves basis independent.

Practical Criterion for Testing Basis Independence

The Tensor Test

To determine whether a proposed array of components represents a genuine basis-independent tensor, it suffices to check its behavior under an arbitrary infinitesimal or finite change of basis and compare against the transformation law; if the array transforms accordingly for every choice of A, it defines a basis-independent tensor, and if it fails for even one choice of A, it does not.

Consequence for Physical and Geometric Laws

This criterion is why equations expressed entirely in tensor form, with all indices properly contracted, are considered to state basis-independent, and hence coordinate-system-independent, relationships, while equations that mix components inconsistently, without balanced upper and lower indices, generally fail to hold once a different basis or coordinate system is adopted.

Diagrammatic Summary

The diagram shows two distinct component lists, arising from two different bases, connected by the transformation law, both describing the identical basis-independent tensor T that exists prior to and independently of either choice of basis.