3.17.4 Tensor Row Vector Coordinate Pairing

Tensor Row Vector Coordinate Pairing is the operation that combines the coordinate components of a covector, arranged as a row, with the coordinate components of a vector, arranged as a column, to produce the scalar value of the covector evaluated on the vector. For a finite-dimensional vector space V with basis {e₁, ..., eₙ} and dual basis {e¹, ..., eⁿ} of V*, a covector ω = ωᵢeⁱ and a vector v = vʲeⱼ are paired according to the rule ω(v) = Σᵢ ωᵢvⁱ, which coincides exactly with the dot product of the row array (ω₁, ..., ωₙ) against the column array (v¹, ..., vⁿ)ᵀ. This pairing is the coordinate-level manifestation of the canonical bilinear evaluation map V* × V → F.

Structure of the Pairing

Bilinearity in Coordinates

The pairing ⟨ω, v⟩ = ω(v) is linear separately in ω and in v. In coordinates this means the pairing distributes over addition and scales properly:

and correspondingly ω(av + bw) = aω(v) + bω(w). Bilinearity is what allows the pairing to be computed term-by-term as a sum of products of matching indices, ωᵢvⁱ, rather than requiring knowledge of the full functional structure of ω at each evaluation.

The Kronecker Delta as the Basic Instance

The most elementary instance of coordinate pairing is the biorthogonality relation defining the dual basis itself: eⁱ(eⱼ) = δⁱⱼ, equal to 1 when i = j and 0 otherwise. Every general pairing ωᵢvⁱ is built from linear combinations of these elementary pairings, since ω(v) = ωᵢeⁱ(vʲeⱼ) = ωᵢvʲeⁱ(eⱼ) = ωᵢvʲδⁱⱼ = ωᵢvⁱ.

Coordinate Pairing and Index Conventions

Einstein Summation

The coordinate pairing is the canonical example motivating the Einstein summation convention, in which a repeated index appearing once as a subscript and once as a superscript is automatically summed without an explicit summation symbol. The expression ωᵢvⁱ is understood to mean Σᵢ ωᵢvⁱ, and the convention is specifically designed so that only pairings between a covariant, lower-indexed quantity and a contravariant, upper-indexed quantity are treated this way, since only such pairings are basis-independent.

Why Mismatched Index Placement Signals an Error

An expression such as ωᵢvᵢ, with both indices written as subscripts, does not represent a coordinate-invariant pairing under a general change of basis, because two lower-indexed arrays do not transform contragrediently to one another. The row vector coordinate pairing convention, by insisting on one upper and one lower repeated index, encodes at the level of notation which contractions are guaranteed to be basis-independent.

Invariance Under Change of Basis

Verification of Invariance

Suppose the basis of V changes by ẽⱼ = Σₖ Aᵏⱼeₖ, so that vector components transform as ṽ = A⁻¹v and covector row components transform as ω̃ = ωA. The coordinate pairing computed in the new basis reproduces the same scalar:

This invariance is the entire justification for pairing a row covector with a column vector using opposite transformation rules: it guarantees that ω(v), the underlying basis-free quantity, is correctly recovered regardless of which coordinate system was used to compute it.

Coordinate Pairing Under Pullback

Pairing After Pulling Back a Covector

If T: V → W is linear with matrix T, and ω is a covector on W, the pulled-back covector Tω pairs with a vector v in V according to (Tω)(v) = ω(Tv), and in coordinates this is (ωT)v, matching the row-vector matrix multiplication description of pullback. The coordinate pairing thus provides the defining computational check that a proposed pullback formula is correct: it must reproduce ω evaluated on the image vector Tv.

Naturality of the Pairing

The coordinate pairing interacts with pullback in a naturality-preserving way: pairing the pulled-back covector Tω with v in V always equals pairing the original covector ω with the pushed-forward vector Tv in W. This identity, ⟨Tω, v⟩_V = ⟨ω, Tv⟩_W, is the coordinate expression of the adjoint relationship between pullback on covectors and the original linear map on vectors.

Coordinate Pairing with Multiple Covectors

Simultaneous Pairing and Matrix Formulation

When several covectors ω₁, ..., ωₖ are stacked as the rows of a k×n matrix Ω, and a vector v is written as a column, the product Ωv produces a column of k scalars, the simultaneous pairing of every ωₐ with v. This generalization underlies how a linear map itself can be regarded as a stack of coordinate pairings, with each row of the map's matrix acting as one covector applied to the input vector.

Pairing and Degenerate Cases

If a covector ω pairs to zero with every vector in a subspace U of V, then ω is said to annihilate U, and the set of all such covectors forms the annihilator U⁰, a subspace of V* whose dimension equals dim(V) minus dim(U). Coordinate pairing thus provides not only individual scalar evaluations but also the basic tool for characterizing entire subspaces through the vanishing of pairings across them.