4.2.1 Tensor Bilinear First Argument Slot

The Tensor Bilinear First Argument Slot defines how a bilinear form acts on its first vector argument, shaping tensor algebra operations and structural interactions.

Tensor Bilinear First Argument Slot is the designated input position of a bilinear map that, when the second argument is held fixed at any chosen value, receives a linear function of that single remaining variable. Given a bilinear map $B : V \times W \to U$ the first argument slot is the coordinate reserved for elements of $V$ , and the defining property of bilinearity guarantees that fixing any $w \in W$ produces a linear map $B (\cdot, w) : V \to U$ acting purely on the first slot. This slotwise behavior is the structural feature that separates a bilinear map from an arbitrary function of two variables, and it is the property exploited throughout tensor algebra when bilinear maps are curried, factored through tensor products, or used to build multilinear forms of higher rank.

Formal Definition of the First Slot

Linearity Condition

For the first argument slot to satisfy bilinearity, two conditions must hold for every fixed $w \in W$ , every $v_{1}, v_{2} \in V$ , and every scalar $α$ from the ground field:

B (v_{1} + v_{2}, w) = B (v_{1}, w) + B (v_{2}, w)

B (α v_{1}, w) = α B (v_{1}, w)

Partial Map Notation

It is standard to denote the resulting partial map by fixing the second slot and writing the induced linear map explicitly as an element of the dual-like space of linear maps from $V$ to $U$ :

B_{w} : V \to U, B_{w} (v) = B (v, w)

The assignment $w \mapsto B_{w}$ is itself linear in $w$ , which is precisely the statement that the two slots of a bilinear map act independently and each one, when isolated, is an ordinary linear map.

Coordinate Description

Matrix Representation

When $V$ and $W$ are finite-dimensional with bases ${e_{i}}$ and ${f_{j}}$ , and $U$ is the scalar field, a bilinear map is represented by a matrix $B = (b_{i j})$ via

B (v, w) = \sum_{i, j} v^{i} b_{i j} w^{j}

The first argument slot corresponds to the row index $i$ : varying $v^{i}$ while $w^{j}$ is fixed produces a linear combination of the row-indexed coefficients, confirming algebraically that the first slot behaves linearly once the second is frozen.

Role of the Column Vector

Equivalently, fixing $w$ and letting $c$ denote the coordinate column of $w$ , the induced linear functional on the first slot is represented by the row vector $v^{T} B c$ , showing that the first slot's linear behavior is literally matrix multiplication acting on the left factor.

Relation to Tensor Product Universal Property

Currying the First Slot

The universal property of the tensor product states that every bilinear map $B : V \times W \to U$ factors uniquely through a linear map $\tilde{B} : V \otimes W \to U$ satisfying $\tilde{B} (v \otimes w) = B (v, w)$ . The first argument slot is precisely the coordinate that gets embedded into the left tensor factor under the canonical map $v \mapsto v \otimes w$ , and its linearity is exactly what makes this embedding well defined for fixed $w$ .

Consequence for Rank-One Tensors

A simple tensor $v \otimes w$ is entirely determined by the pair of slot values, and the first slot's linearity guarantees that

(v_{1} + v_{2}) \otimes w = v_{1} \otimes w + v_{2} \otimes w

holds identically inside the tensor product space, which is the abstract expression of the first slot's linear behavior.

Independence from the Second Slot

Why the Two Slots Do Not Interact

Bilinearity does not require any relationship between how the first slot behaves and how the second slot behaves; each is linear in isolation while the other is frozen, but jointly the map is generally not linear on $V \times W$ as a single vector space, since $B (α v, α w) = α^{2} B (v, w)$ rather than $α B (v, w)$ . This distinguishes slotwise linearity, which the first argument slot exhibits, from full linearity over the product space.

Interaction with the Second Argument Slot

Because the first slot and the second slot are each linear independently, the map $B$ is fully specified by its values on pairs of basis vectors drawn one from each slot, and every bilinear map on finite-dimensional spaces reduces to the matrix description given above. This is the sense in which the first argument slot, together with the second, generates the entire structure of the bilinear map from finitely many scalar data.

Significance in Building Higher Tensors

Extension to Multilinear Maps

The notion of an argument slot generalizes directly to maps of three or more variables, where each slot independently satisfies a linearity condition analogous to the one described here. The first argument slot of a bilinear map is the base case of this pattern and is the structural template used when defining trilinear and higher multilinear maps used to construct tensors of arbitrary rank.

Practical Use in Contraction

When a bilinear map is contracted against a fixed vector to produce a linear functional, it is conventionally the first slot that is held variable while a specific element is substituted into the second, or vice versa; the terminology "first argument slot" exists precisely to make this substitution order and its resulting linear map unambiguous in multi-step tensor computations.