3.17.2 Tensor Row Vector Basis Dependence

Tensor Row Vector Basis Dependence is the property by which the numerical entries of a covector's row representation change when the basis of the underlying vector space is changed, even though the covector itself, as an abstract linear functional, remains the same invariant object. Given a vector space V with basis {e₁, ..., eₙ} and dual basis {e¹, ..., eⁿ}, a covector ω = ωᵢeⁱ has row form (ω₁, ..., ωₙ) relative to that basis; adopting a different basis for V induces a different dual basis and, in general, produces an entirely different array of numbers representing the same functional ω. Basis dependence is not a defect but a structural feature that must be tracked correctly whenever coordinates are used to compute with covectors.

Source of the Dependence

The Dual Basis Is Not Free-Standing

The dual basis {eⁱ} is defined only relative to a chosen basis {eⱼ} of V through the biorthogonality condition eⁱ(eⱼ) = δⁱⱼ. There is no dual basis attached to V* independent of a prior choice of basis for V. Consequently, the row form of ω, which lists ω's coefficients with respect to {eⁱ}, inherits basis dependence directly from this definitional link.

Change of Basis on V

Suppose a new basis is defined by ẽⱼ = Σₖ Aᵏⱼeₖ for an invertible matrix A = (Aᵏⱼ). The associated dual basis transforms by the inverse transpose of A, so that the new row components of ω satisfy

meaning ω̃ⱼ = Σᵢ ωᵢAⁱⱼ. This is a right-multiplication of the row array by A, distinct from the left-multiplication by A⁻¹ that governs how column vectors transform.

Invariance of the Underlying Functional

The Pairing Remains Fixed

Although the numerical row array changes with the basis, the scalar produced by evaluating ω on any fixed vector v does not. If v has components vʲ relative to {eⱼ} and ṽᵏ relative to {ẽⱼ}, with ṽ = A⁻¹v, then

The cancellation of A and A⁻¹ is the precise mechanism ensuring that basis dependence in the row form of ω and the column form of v exactly compensate, leaving the abstract pairing ω(v) invariant. This cancellation is the defining feature separating covariant from contravariant transformation behavior.

Contragredience as the Organizing Principle

Because covector row components transform by A while vector column components transform by A⁻¹, the two transformation laws are called contragredient to one another. Basis dependence of the row form is therefore not arbitrary noise but follows a precise, predictable rule dictated entirely by the change-of-basis matrix used for V.

Consequences for Computation

Coordinates Are Not Intrinsic Data

Any statement about a covector expressed purely in terms of its row components, such as "ω has a zero in its third entry," is meaningful only relative to a specified basis. The same covector may have no zero entries in a different basis. Care must be taken in applications, such as identifying when a covector annihilates a particular vector, to either fix a single basis throughout a computation or explicitly track how both the covector and the vectors involved transform together.

Basis-Independent Quantities Built from Row Vectors

Certain quantities constructed from the row form remain invariant despite the basis dependence of the individual entries. The scalar ω(v) is one example; the rank of a set of covectors, meaning the dimension of their span in V*, is another, since rank is unaffected by the invertible linear substitution induced by a change of basis. Recognizing which derived quantities are invariant and which are merely coordinate artifacts is essential to correctly interpreting row vector data.

Relation to Choice of Dual Basis Versus Choice of Metric

Two Independent Sources of Apparent Change

Basis dependence of the row form must be distinguished from the separate issue of choosing an inner product to identify V with V*. Changing the basis of V, with no inner product involved at all, already changes the row components of every covector according to the contragredient rule. Introducing or changing an inner product is a separate operation that affects how covectors are associated with vectors, but not, by itself, how a covector's row components transform under a change of basis of V.

Orthonormal Bases and Apparent Simplification

If attention is restricted to orthonormal bases relative to some fixed inner product, the transformation matrix A relating two such bases is orthogonal, so A⁻¹ = Aᵀ, and the row form of a covector transforms by the same matrix, up to transpose, that governs the associated vector under the musical isomorphism. This special coincidence, valid only for orthonormal-to-orthonormal changes, can obscure the general basis dependence of row vectors and should not be mistaken for a universal simplification.

Row Vector Basis Dependence Under Linear Maps

Interaction with Pullback

When a covector ω on a space W is pulled back along a linear map T: V → W to form Tω on V, the row form of Tω, namely ωT, depends jointly on the basis chosen for V, through the matrix representation of T, and the basis chosen for W, through the row components of ω itself. Changing either basis independently changes the row array of T*ω, so basis dependence propagates through composite constructions exactly as it does for a single covector, and must be tracked at each stage of a multi-map computation.