3.17 Tensor Row Vector Representation

Tensor Row Vector Representation is the convention by which a covector, that is, an element of the dual space of a finite-dimensional vector space, is written as a one-row matrix of components acting on column vectors by ordinary matrix multiplication. Given a vector space V of dimension n with basis e₁, ..., eₙ, and its dual space V* with dual basis e¹, ..., eⁿ satisfying eⁱ(eⱼ) = δⁱⱼ, any covector ω in V* can be expanded as ω = ωᵢeⁱ, where the coefficients ωᵢ are the components of ω relative to the dual basis. Representing ω as a row vector [ω₁ ω₂ ... ωₙ] and representing a vector v = vʲeⱼ as a column vector with entries vʲ makes the pairing ω(v) coincide exactly with the matrix product of a 1×n row by an n×1 column, producing a single scalar.

Foundational Structure

Duality and the Pairing Operation

The dual space V* consists of all linear functionals V → F, where F is the underlying field, typically the real or complex numbers. The defining feature of the row vector representation is that it encodes this abstract linear functional as a concrete numerical array whose action on any vector is computed by the standard row-times-column matrix product.

Extraction of Components

Because the dual basis satisfies eⁱ(eⱼ) = δⁱⱼ, the i-th component of a covector ω in row form is recovered by evaluating ω on the i-th basis vector: ωᵢ = ω(eᵢ). This mirrors how the j-th component of a vector v in column form is recovered by applying the j-th dual basis covector: vʲ = eʲ(v). The row and column conventions are therefore not arbitrary notational choices but reflect the genuinely different transformation behavior of vectors and covectors.

Relationship to Vectors and Transpose

Row Versus Column as a Structural Distinction

In many introductory treatments, especially in physics and engineering, row vectors and column vectors are presented as mere transposes of one another within a single vector space equipped with an inner product. The tensor-algebraic viewpoint is more precise: a row vector is not simply "a vector written sideways" but a genuinely different type of object, a covector, living in a different space V* that is only isomorphic to V once an inner product or other non-canonical identification is chosen. The row vector representation is thus the natural, basis-dependent shadow of the abstract dual space, not an accident of typography.

The Canonical Pairing as Matrix Multiplication

The evaluation map V* × V → F, (ω, v) ↦ ω(v), is bilinear and, once bases are fixed, is realized exactly as 1×n times n×1 matrix multiplication. This is the deepest justification for the row/column convention: it allows the abstract, basis-free pairing between a dual space and its underlying space to be computed using nothing more than the ordinary rules of matrix algebra.

Transformation Under Change of Basis

Contragredient Behavior

If the basis of V is changed by an invertible matrix A, so that new basis vectors are ẽⱼ = Aᵏⱼeₖ, then vector components transform by the inverse of A, while covector components transform by A itself acting on the row from the right, or equivalently by Aᵀ acting on the column form of ω transposed. Explicitly, if ω̃ denotes the row vector of components in the new dual basis, then

while for vectors, ṽ = A⁻¹v. This opposite, contragredient transformation law is precisely what guarantees that the scalar ω(v) = ω̃(ṽ) is invariant under change of basis, since ωA A⁻¹v = ωv. The row vector representation makes this cancellation transparent as ordinary matrix associativity.

Index Placement Convention

Consistent with this transformation law, covector components are conventionally written with a lower index, ωᵢ, while vector components carry an upper index, vʲ. The row vector representation is the notational vehicle that keeps this distinction visible: lower-indexed quantities are arranged horizontally, upper-indexed quantities vertically, so that a repeated index appearing once up and once down, summed according to the Einstein convention, automatically corresponds to a legal row-times-column contraction.

Computational Consequences

Composition with Linear Maps

If T: V → W is a linear map represented by a matrix, and ω is a covector on W represented as a row vector, then the pullback covector Tω on V, defined by (Tω)(v) = ω(T(v)), is represented in row form by the matrix product ω T, where T is the matrix of T acting on column vectors. This shows that pulling back a covector along a linear map corresponds to right-multiplying its row representation by the map's matrix, in contrast to pushing a vector forward, which corresponds to left-multiplying a column vector by the same matrix.

Row Vectors in Numerical Practice

In computational linear algebra, storing covectors as row arrays rather than column arrays is more than cosmetic: it allows a single evaluation ωv to be computed with one dot-product pass, and it allows batches of vectors, stored as the columns of a matrix, to be paired with a fixed covector via a single row-times-matrix multiplication, producing a row of scalars, one per input vector. This efficiency is a direct, practical consequence of respecting the dual-space structure rather than collapsing rows and columns into a single undifferentiated notion of "vector."

Limitations of the Representation

Basis Dependence

The row vector representation is tied to a specific choice of basis for V and the induced dual basis for V*. Changing the basis changes the numerical entries of the row vector even though the underlying linear functional is unchanged. The representation is therefore a coordinate description of an intrinsically coordinate-free object, useful for computation but not itself the invariant content of the covector.

Dependence on an Ambient Inner Product for Identification with Column Vectors

When authors informally identify a covector's row form with the transpose of "the same" vector's column form, they are implicitly invoking a metric or inner product to construct an isomorphism between V and V*. Without such an additional structure, there is no canonical way to turn a row vector into a column vector of the same space; the row vector representation of a covector and the column vector representation of a vector remain conceptually distinct even when a chosen inner product makes them numerically interchangeable.