3.17.1 Tensor Covector Row Form

Tensor Covector Row Form is the explicit rendering of an element of a dual vector space as a single-row array of scalar components, arranged so that the linear pairing between the covector and any vector reduces to ordinary matrix multiplication of the row by a column. If V is a finite-dimensional vector space over a field F with basis {e₁, ..., eₙ} and dual basis {e¹, ..., eⁿ} for V* satisfying eⁱ(eⱼ) = δⁱⱼ, then a covector ω = ωᵢeⁱ is displayed in row form as the 1×n array (ω₁, ω₂, ..., ωₙ). This form is the standard computational and typographic device for distinguishing covariant objects from the contravariant column arrays used for vectors.

Definition and Basic Mechanics

Constructing the Row Form

Each component ωᵢ is obtained by applying the covector to the i-th basis vector, ωᵢ = ω(eᵢ). Collecting these n scalars in order produces the row array. Because the map ω ↦ (ω₁, ..., ωₙ) is linear and bijective once the basis is fixed, the row form is a faithful coordinate representation of ω, though, as with any coordinate representation, it depends on the chosen basis.

The Pairing as Matrix Product

For a vector v = vʲeⱼ written as the column array (v¹, ..., vⁿ)ᵀ, the evaluation ω(v) is computed as

where the right-hand side denotes literal 1×n by n×1 matrix multiplication. The row form is precisely the arrangement of ω's components that makes this identity hold using unmodified matrix arithmetic, with no transposition or reordering required.

Distinguishing Row Form from Mere Transposition

Covectors Are Not Transposed Vectors in General

It is tempting, particularly in Euclidean settings, to regard the row form of a covector as simply the transpose of a corresponding column vector. This identification is only valid once an inner product ⟨ , ⟩ on V has been fixed, inducing the musical isomorphism V → V*, v ↦ v♭ = ⟨v, ·⟩. Without such a choice, V* has no canonical basis-independent identification with V, and the row form must be understood as the coordinate expression of a genuinely separate object: a linear functional, not a repositioned vector.

Behavior Under Orthonormal Bases

When the chosen basis of V is orthonormal with respect to a fixed inner product, the components of v♭ in row form numerically coincide with the components of v in column form. This coincidence is a special convenience of orthonormal coordinates and disappears under a general change of basis or in the absence of a metric, reinforcing that the row form is fundamentally a dual-space object.

Transformation Law

Contragredient Rule for Row Components

Under a change of basis of V given by ẽⱼ = Σₖ Aᵏⱼeₖ for an invertible matrix A, the row form of a covector transforms as

that is, the new row is obtained by right-multiplying the old row by A. This contrasts with column vectors, whose components transform by ṽ = A⁻¹v. The opposite direction of these two rules, called contragredience, ensures the scalar pairing ω(v) is basis-independent, since ω̃ṽ = ωAA⁻¹v = ωv.

Consistency with Lower-Index Notation

The row form is the notational counterpart of writing covector components with a subscript, ωᵢ, as opposed to the superscript vʲ used for vector components. Placing lower-indexed quantities horizontally and upper-indexed quantities vertically is a mnemonic device ensuring that any expression respecting the Einstein summation convention, with one repeated upper and one repeated lower index, automatically corresponds to a well-formed row-times-column contraction.

Row Form in Composite Operations

Row Vectors and Linear Map Pullback

Given a linear map T: V → W with matrix representation T acting on column vectors, and a covector ω on W in row form, the pulled-back covector Tω on V, defined by (Tω)(v) = ω(Tv), has row form equal to the matrix product ωT. The pullback operation on covectors is thus realized, in row form, as right-multiplication by the map's matrix, which is the mirror image of how vectors are pushed forward by left-multiplication.

Row Form and Rank-One Tensors

The outer product of a row covector ω with itself or with another row covector η produces components of a rank-two covariant tensor, ωᵢηⱼ, which can be arranged as an n×n matrix. More generally, row form provides the building block for representing symmetric bilinear forms, such as metric tensors, and antisymmetric forms, such as differential two-forms, once multiple covectors are combined through tensor or wedge products.

Practical and Symbolic Considerations

Why the Row Convention Is Preserved in Practice

Retaining the row form for covectors, rather than collapsing it into column notation via transposition, keeps formulas that mix covariant and contravariant quantities visually and algebraically self-checking: a legitimate contraction always pairs a row with a column, and any expression violating this pattern signals either a notational error or an implicit and often unjustified metric identification.

Row Form Under Composition of Several Maps

If covectors are pulled back through a chain of linear maps T₁, T₂, ..., Tₖ, the row form composes by successive right-multiplication in the same order the maps are applied to vectors, namely (T₁T₂⋯Tₖ)*ω = ω(T₁T₂⋯Tₖ), which reverses to η = ω T₁ then η T₂ and so on only when tracked carefully; the row form thus provides a direct, matrix-algebraic bookkeeping tool for chains of pullback operations without needing to reformulate the underlying linear algebra.