10.1.2 Tensor Change of Basis Component Scope

Understanding how tensor components transform under basis changes, focusing on their scope and mathematical behavior in linear algebra.

Tensor Change of Basis Component Scope is the delineation of which specific component types, index positions, and tensor ranks are governed by the change-of-basis transformation rule, clarifying exactly which numerical quantities attached to a tensor are subject to this rule and which quantities, despite superficial resemblance, are not.

Component Types Covered

Contravariant Components at Any Rank

The scope includes every contravariant component slot of a tensor, from a single vector's components up through arbitrarily many upper indices on a higher-rank tensor, each transforming with its own factor of the inverse change-of-basis matrix.

T^{i_{1} \dots i_{p}}' = {(A^{- 1})}_{j_{1}}^{i_{1}} \dots {(A^{- 1})}_{j_{p}}^{i_{p}} T^{j_{1} \dots j_{p}}

Covariant Components at Any Rank

Likewise, the scope includes every covariant component slot, each transforming with its own factor of the change-of-basis matrix directly, with no restriction on how many such indices a tensor carries.

T_{j_{1} \dots j_{q}}' = A_{j_{1}}^{l_{1}} \dots A_{j_{q}}^{l_{q}} T_{l_{1} \dots l_{q}}

Mixed Components Combining Both Types

The scope covers mixed tensors with any combination of upper and lower indices, applying the appropriate factor to each slot independently regardless of the order in which upper and lower indices appear.

T_{j}^{i}

Component-Level Exclusions

Scalars Are Trivially in Scope but Unaffected

A scalar, being a rank-zero tensor with no indices at all, is technically within the component scope but is left completely unchanged by any basis transformation, since there is no index to apply a factor of (A) or (A^{-1}) to; this is the degenerate boundary case of the scope rather than an exception to it.

ϕ' = ϕ

Numerical Labels That Are Not Tensor Components

Indices used purely as labels, such as an index distinguishing separate tensors in a list rather than denoting a tensorial slot, fall outside this component scope entirely, since they do not correspond to a basis direction and therefore do not transform under a change of basis at all.

Components of Non-Tensorial Objects

Arrays such as Christoffel symbols, which carry indices but do not satisfy the tensor transformation law due to an extra inhomogeneous term, are excluded from this component scope even though their index notation looks identical to that of genuine tensor components.

Rank as the Organizing Boundary

Scope Scales With, but Is Not Limited By, Rank

The component scope makes no distinction based on how large (p) and (q) are; the same rule that governs a single vector's component extends without modification to a tensor of any finite rank, with the total number of transformation factors simply growing to match the number of indices present.

p + q = total number of transformation factors applied

Visual Illustration

Why This Component-Level Precision Matters

Precisely scoping which components are governed by the change-of-basis rule prevents the rule from being misapplied to superficially similar but fundamentally different indexed quantities. By confirming that the scope covers every contravariant and covariant slot uniformly, at any rank, while explicitly excluding non-tensorial indexed objects and non-basis-related labels, this scope keeps the transformation rule both fully general within genuine tensors and correctly bounded against objects that merely resemble them.