7.3.1 Tensor Component Basis Dependence

Tensor Component Basis Dependence explores how tensor components change with basis transformations, revealing intrinsic geometric relationships in multilinear algebra.

Tensor Component Basis Dependence is the property by which the numerical entries of a tensor's component array are defined only relative to a chosen basis of the underlying vector space and its dual, so that no single component carries meaning on its own, apart from stating which basis vectors and covectors it is measured against.

Definition and Scope

Components as Expansion Coefficients

A tensor's components are the coefficients obtained when the tensor is expanded in terms of basis vectors (e_i) of the vector space and basis covectors (e^j) of its dual. A ((1,1)) tensor (T) is written as

T = T_{j}^{i} e_{i} \otimes e^{j}

and the numbers (T^i_{\ j}) are precisely the coefficients required to reconstruct (T) from this particular pair of bases; a different choice of (e_i) and (e^j) produces a different set of coefficients representing the same (T).

Basis Versus Coordinate System

Basis dependence refers specifically to the choice of vectors spanning the space at a point, whereas a coordinate system additionally assigns those vectors smoothly across a region, as in curvilinear coordinates. Basis dependence is the more elementary notion: even at a single point, before any coordinate patch is introduced, the components of a tensor already depend on which basis vectors were selected to measure it against.

Structural Properties

The Change-of-Basis Rule

If a new basis (e_i') is related to the old one by (e_i' = A_i^{\ k} e_k), the components of a tensor transform oppositely for upper and lower indices:

T_{j}^{'} = {(A^{- 1})}_{k}^{i} A_{j}^{l} T_{l}^{k}

so that upper-index components transform with the inverse of the matrix relating the bases, while lower-index components transform with the matrix itself, keeping the tensor (T) fixed while its components shift.

Orthonormal Versus Non-Orthonormal Bases

The specific numerical values a tensor's components take can differ sharply depending on whether the chosen basis is orthonormal or not. In a non-orthonormal basis, distinct roles typically played by upper and lower indices, such as the values of a metric tensor, become visibly asymmetric, whereas an orthonormal basis often collapses this distinction, making upper and lower components numerically identical for a metric tensor equal to the identity.

Basis-Independent Quantities

Not every quantity derived from a tensor is basis dependent. Scalars formed by fully contracting all of a tensor's indices, such as a trace or a norm built from a metric, are constructed exactly so as to be invariant under any change of basis, providing a basis-independent handle on an otherwise basis-dependent object.

Role Within Tensor Algebra

Consistency Requirement for Tensor Operations

Two sets of components can only be combined through addition, contraction, or comparison if they are expressed in the same basis; components computed in different bases must first be related through the change-of-basis rule before any such operation is meaningful, since a tensor's identity, not its numerical appearance, is what operations act on.

Practical Implications

In applied settings, basis dependence explains why the same physical tensor, such as an inertia tensor or a metric, can appear with a simple, diagonal component structure in a basis aligned with the object's natural symmetry axes, and with a dense, non-diagonal structure in an arbitrarily chosen basis, without the underlying physical quantity changing at all.