10.5.3 Tensor Forward Basis Matrix Direction

Tensor Forward Basis Matrix Direction defines how basis vectors transform under linear mappings, establishing a framework for tensor operations in multi-dimensional spaces.

Tensor Forward Basis Matrix Direction is the specific assignment, fixed by the forward rule, of which matrix, (A) or its inverse (A^{-1}), points from source to target for the basis vectors versus for each type of tensor component, clarifying that a single notion of "forward" produces opposite matrix choices depending on what is being transformed.

Direction for the Basis Itself

A Points From Source to Target for Basis Vectors

For the basis vectors, the forward direction is unambiguous: (A) is defined precisely as the matrix carrying the source basis vectors to the target basis vectors, so (A) itself is, by definition, the forward-direction matrix for the basis.

e_{i}' = A_{i}^{j} e_{j}

Direction for Contravariant Components

A Inverse Points Forward for Contravariant Components

For contravariant components, the matrix pointing in the forward direction, from source components to target components, is not (A) but (A^{-1}), since preserving the underlying vector forces the components to compensate oppositely to how the basis vectors themselves transform.

v^{i}' = {(A^{- 1})}_{j}^{i} v^{j}

This is the single most important direction fact to internalize: despite (A) being labeled the forward matrix for the basis, the forward matrix for contravariant components is its inverse, not (A) itself.

Direction for Covariant Components

A Points Forward for Covariant Components

For covariant components, the forward direction matches the basis vectors' own direction, with (A) applied directly rather than its inverse.

ω_{i}' = A_{i}^{j} ω_{j}

Covariant components thus share their forward-direction matrix with the basis vectors, while contravariant components use the opposite matrix, a pairing pattern that holds without exception across every tensor of any rank.

Summary of Direction Assignment by Object Type

A Single Table of Correspondences

Collecting these results together, the forward direction assigns (A) to the basis vectors and to every covariant index, and assigns (A^{-1}) to every contravariant index, with a mixed tensor requiring the appropriate combination of both matrices simultaneously depending on how many indices of each type it carries.

\begin{matrix} basis vectors & \to & A \\ covariant components & \to & A \\ contravariant components & \to & A^{- 1} \end{matrix}

Why the Direction Splits This Way

Consequence of the Object Preservation Requirement

This apparent asymmetry, where the same forward direction uses (A) for some objects and (A^{-1}) for others, is not an arbitrary convention but a direct, forced consequence of the object preservation requirement: substituting either choice the wrong way into the invariance identity fails to reproduce the same tensor, confirming that this specific direction assignment is the only one consistent with preserving the underlying object.

v^{i} e_{i} = v^{i}' e_{i}'

Visual Illustration

Why Getting This Direction Assignment Right Matters

Misapplying the matrix direction, using (A) where (A^{-1}) is required for a contravariant index, or the reverse, is one of the most common practical errors in basis transformation, precisely because it is tempting to assume every object transforms in the same direction as the basis vectors. Internalizing this precise, non-uniform direction assignment is what allows the forward rule to be applied correctly and confidently to any tensor, regardless of how many indices of each type it carries.