3.14 Tensor Covector Transformation Behavior

Tensor Covector Transformation Behavior is the specific rule describing how the components of a covector change when the basis of the underlying vector space is replaced by a different basis, characterized by the components transforming using the same change-of-basis matrix that relates the old and new bases, rather than its inverse. This behavior is what distinguishes covectors from vectors at the level of components, and it is the property that earns covector indices the label "covariant," since their transformation follows, or co-varies with, the transformation of the basis itself.

Setting Up a Change of Basis

Old and New Bases

Let V be a finite-dimensional vector space with an old basis e_1, ..., e_n and a new basis f_1, ..., f_n. The new basis vectors are expressed in terms of the old ones by a change-of-basis matrix A, with entries A^j_i, according to:

Corresponding Dual Bases

Each basis of V induces a dual basis of V*: the old basis e_1, ..., e_n induces e^1, ..., e^n, and the new basis f_1, ..., f_n induces f^1, ..., f^n, each characterized by returning 1 when paired with the matching basis vector and 0 otherwise.

Derivation of the Covector Transformation Rule

Expressing the New Dual Basis in the Old One

To find how covector components transform, it is first necessary to express the new dual basis f^1, ..., f^n in terms of the old dual basis e^1, ..., e^n. Requiring f^i(f_j) = δ^i_j and substituting the expression for f_j in terms of the e_k leads to:

so the dual basis transforms using the inverse matrix A^{-1}, in contrast to the basis vectors themselves, which transform using A directly.

Components of a Fixed Covector

A covector φ in V* has components φ_i relative to the old dual basis and components φ'_i relative to the new dual basis, defined by φ = φ_i e^i = φ'_i f^i. Substituting the relation between f^i and the e^k and comparing coefficients yields the transformation law for the components themselves:

Here the covector components transform using the matrix A directly, the same matrix used to express the new basis vectors f_i in terms of the old ones e_j.

Interpreting the Result

"Co-varying" with the Basis

The name covariant reflects the fact that the components φ_i transform with the same matrix A that describes how the basis vectors themselves change. When the basis vectors are transformed by A, the covector components are transformed by that same A, so the components are said to vary together with, or co-vary with, the basis, rather than against it.

Comparison with Vector (Contravariant) Components

The components v^i of an ordinary vector v = v^i e_i = v'^i f_i transform according to a different rule, using the inverse matrix:

The opposite behavior of the two transformation rules, A for covector components and A^{-1} for vector components, is precisely what is needed for the evaluation pairing φ(v) to remain unchanged regardless of which basis is used to compute it.

Invariance of the Pairing Under the Transformation Rule

Why the Two Rules Must Be Inverse to Each Other

The requirement that φ(v) be a basis-independent scalar places a strong constraint on how covector and vector components can transform. If vector components transform by some matrix B under a change of basis, then covector components must transform by (B^{-1})^T or an equivalent inverse-type matrix so that the sum Σ φ_i v^i produces the same numerical value in every basis. The specific choice B = A^{-1} for vectors and the corresponding A for covectors, as derived above, is exactly the pairing of transformation rules that achieves this invariance.

Verifying Invariance Directly

Computing the pairing in the new basis and substituting both transformation laws confirms the expected cancellation:

so the numerical value of the pairing computed from primed components matches the value computed from unprimed components, confirming that φ(v) does not depend on the basis chosen to express φ and v in coordinates.

Extension to General Tensor Indices

Covariant Indices in a General Tensor

The transformation behavior described for a single covector generalizes directly to each lower, covariant index of an arbitrary tensor. A tensor with several lower indices has each one transform with a separate factor of the matrix A, mirroring the single-index rule for covectors applied independently to each covariant slot.

Consistency Across Mixed Tensors

In a tensor with both upper and lower indices, the upper indices transform with A^{-1}, following the vector rule, while the lower indices transform with A, following the covector rule described here. This combination of transformation behaviors is what allows a mixed tensor to be simultaneously interpreted as built from copies of V and copies of V*, with each factor transforming according to its own type.

Diagrammatic Summary

The diagram places covector transformation using A above and vector transformation using A^{-1} below, side by side, to make visible the opposite direction in which the two types of components respond to the same change-of-basis matrix.