10.1 Tensor Change of Basis Scope

Tensor Change of Basis Scope is the delineation of exactly which objects, operations, and transformation rules fall under the study of how tensors and their components behave when the underlying basis of a vector space is replaced by another, as distinct from related topics such as coordinate charts on manifolds or transformations of the vector space itself.

What Falls Inside the Scope

Components of Vectors, Covectors, and General Tensors

The scope covers the transformation rules for the components of any tensor of any type ((p, q)), specifying exactly how each contravariant and covariant index responds to a change from one basis to another.

This includes vectors and covectors as the simplest cases, matrices as rank-two tensors, and general higher-rank tensors built from tensor products of the basic cases.

The Change-of-Basis Matrix and Its Inverse

Central to the scope is the invertible matrix relating an old basis to a new one, along with its inverse, since every transformation rule for tensor components is expressed in terms of one or the other.

Basis-Dependent Quantities Built From Tensors

The scope extends to quantities derived directly from tensors using only basis-dependent operations, such as raising and lowering indices with a metric tensor, or contracting indices, since these operations must be shown to respect the change-of-basis rules for the overall scope to be self-consistent.

What Falls Outside the Scope

Coordinate Charts on Curved Manifolds

While closely related, the behavior of coordinate basis vectors that vary from point to point across a curved manifold, and the associated machinery of covariant differentiation, lies outside the narrower scope of tensor change of basis at a single point or in a single vector space; that broader subject belongs to differential geometry and tensor calculus rather than to basis change within a fixed vector space.

Transformations of the Vector Space Itself

A change of basis leaves the vector space and every vector in it completely unchanged; it only changes the labels used to describe them. This is distinct from an active transformation, such as a rotation applied physically to a vector, which does change the vector itself and falls outside the scope of change of basis proper, even though the same matrices may appear in both contexts.

Nonlinear Reparametrizations Without a Linear Basis Change

Scope is restricted to changes that can be described, at least locally, by a linear invertible matrix acting on basis vectors; wholly nonlinear reparametrizations of a space that are not tied to any underlying linear basis change, such as arbitrary smooth relabelings without a corresponding vector space basis, sit outside this scope and are addressed under coordinate transformation theory instead.

Boundaries of Applicability Within the Scope

Restriction to Invertible, Same-Dimension Bases

Within scope, only transformations between two bases of the same finite dimension, related by an invertible matrix, are considered; changes involving a different number of basis vectors, or non-invertible maps, do not constitute a valid change of basis and are excluded.

Visual Illustration

Why the Scope Is Drawn This Way

Fixing the scope of tensor change of basis to component transformation within a single, fixed vector space keeps the subject self-contained and precisely tractable using linear algebra alone. Related but distinct subjects, such as covariant differentiation on manifolds or active geometric transformations, build directly on this foundation but require additional structure beyond what a simple change-of-basis matrix provides, which is why they are treated as separate topics rather than folded into this scope.