3.17.3 Tensor Row Vector Matrix Multiplication

Tensor Row Vector Matrix Multiplication is the operation of combining a row-form covector with a matrix, either representing a linear map or a change of basis, by ordinary matrix multiplication applied on the right, producing a new row vector that represents either a transformed covector or the same covector expressed in new coordinates. If ω = (ω₁, ..., ωₙ) is a covector in row form and M is an n×n matrix, the product ωM is again a 1×n row array, obtained by the standard rule of multiplying a row by a matrix, and this single algebraic operation underlies pullback of covectors, change-of-basis transformation of covector components, and composition of successive linear substitutions.

The Mechanics of Right Multiplication

Definition of the Product

For ω a 1×n row vector and M an n×p matrix, the product ωM is the 1×p row vector whose j-th entry is

This is exactly the contraction of ω's index against the first index of M, and it is well defined precisely because the number of columns in ω, treated as a 1×n matrix, matches the number of rows in M.

Why Multiplication Occurs on the Right

Row vectors multiply matrices on the right because a covector consumes a vector, and a matrix M acting on a column vector v produces Mv, so that a covector applied afterward must appear to the left of M in order to receive Mv as its argument: ω(Mv) = (ωM)v. Attempting to multiply on the left, Mω, is generally not even defined unless M happens to be square and the shapes coincide, and even then it does not correspond to any natural operation on the covector ω.

Row Vector Multiplication as Pullback

The Pullback Formula

If T: V → W is a linear map with matrix representation T relative to chosen bases, acting on column vectors of V to produce column vectors of W, and ω is a covector on W in row form, the pullback covector Tω on V is defined by (Tω)(v) = ω(Tv) for every v in V. In row-vector matrix language this is exactly

so pullback of a covector along T is computed by right-multiplying its row form by the very same matrix T that pushes vectors forward, with no transposition needed because the row/column convention already places the indices in the correct contracting positions.

Composability of Pullback Multiplication

If S: U → V and T: V → W are linear maps with matrices S and T, then the pullback along the composite T∘S satisfies (T∘S)*ω = ω(TS) = (ωT)S, which shows that row-vector matrix multiplication is associative in a way that exactly mirrors, and is dual to, the composition rule (T∘S)(u) = T(S(u)) for the maps themselves acting on vectors. Pulling back first through T and then through S in row notation reproduces the same result as pulling back through the single composite map TS.

Row Vector Multiplication as Change of Basis

Transformation of Components

When M represents not a map between different spaces but a change of basis on a single space V, given by ẽⱼ = Σₖ Mᵏⱼeₖ, the row components of a fixed covector ω transform by the same right-multiplication rule, ω̃ = ωM, expressing the new components in terms of the old. This is formally identical to the pullback formula, reflecting the fact that a change of basis can itself be regarded as the pullback of covectors along the identity map expressed in two different coordinate systems.

Contrast with Column Vector Transformation

Column vectors transform under the same basis change by left-multiplication with the inverse, v ↦ M⁻¹v, so that vector and covector components transform by inverse and non-inverse forms of the same matrix, applied on opposite sides. This contragredient pairing of left-multiplication-by-inverse for columns and right-multiplication-by-original for rows is precisely what keeps the scalar pairing ω(v) invariant under any change of basis.

Rank and Kernel Behavior Under Multiplication

Effect on the Space of Achievable Covectors

The set of all row vectors obtainable as ωT, as ω ranges over all covectors on W, forms the row space of the matrix T, which has dimension equal to the rank of T. If T is not surjective, the pulled-back covectors Tω do not exhaust the full dual space V, meaning certain covectors on V can never arise as pullbacks under T; if T is not injective, distinct covectors ω on W may or may not produce distinct row vectors ωT depending on whether they differ on the image of T.

Multiplication and the Kernel of T

A row vector ω satisfies ωT = 0 exactly when ω vanishes on the entire image of T, since (ωT)(u) is really shorthand for ω(Tu) for every relevant u, so the covectors annihilated by right multiplication with T correspond precisely to those vanishing on im(T), forming a subspace of W* whose dimension equals dim(W) minus the rank of T.

Practical Chains of Multiplication

Successive Row Vector Updates

In applications involving repeated coordinate changes or a sequence of linear stages, such as successive layers of a linear transformation pipeline, a single covector's row form is updated by repeatedly right-multiplying by each stage's matrix in the order the stages are applied to vectors, ω_final = ω₀M₁M₂⋯Mₖ, allowing the cumulative effect of many transformations to be tracked with one growing chain of matrix products rather than recomputing the pairing from scratch at every stage.