4.3.1 Tensor Trilinear First Argument Slot

The Tensor Trilinear First Argument Slot refers to the first input in a trilinear operation, defining how tensors interact in multi-linear algebra.

Tensor Trilinear First Argument Slot is the designated input position of a trilinear map that, when the second and third arguments are both held fixed at any chosen values, receives a linear function of that single remaining variable. Given a trilinear map $T : V \times W \times X \to U$ the first argument slot is the coordinate reserved for elements of $V$ , and trilinearity guarantees that fixing any $w \in W$ and $x \in X$ produces a linear map $T (\cdot, w, x) : V \to U$ acting purely on the first slot. Isolating this slot is the standard technique for reducing questions about trilinear maps to the already well-understood theory of linear maps.

Formal Statement

The Linearity Condition

For every fixed pair $w \in W$ , $x \in X$ , every $v_{1}, v_{2} \in V$ , and every scalar $α$ , the first argument slot satisfies

T (v_{1} + α v_{2}, w, x) = T (v_{1}, w, x) + α T (v_{2}, w, x)

This condition depends on both other slots being fixed simultaneously; freezing only one of them and letting the other vary along with the first slot would not, in general, produce a linear expression, since the map is only guaranteed to be linear in exactly one slot at a time.

Notation for the Frozen Partial Map

It is standard to write the partial map obtained by freezing the second and third slots as

T_{w, x} : V \to U, T_{w, x} (v) = T (v, w, x)

with the double subscript signaling that two slots, not one, have been fixed to produce this linear map.

Coordinate Description

Index Position in the Structure Constants

In coordinates, if $c_{i j k}^{l}$ denotes the components of $T$ relative to chosen bases, the first argument slot corresponds to the index $i$ . Fixing $j$ and $k$ to the coordinates of specific vectors $w$ and $x$ leaves an expression linear in $v^{i}$ :

T (v, w, x) = \sum_{i} v^{i} (\sum_{j, k} c_{i j k}^{l} w^{j} x^{k}) u_{l}

The quantity in parentheses is a fixed coefficient once $w$ and $x$ are chosen, confirming linearity of the first slot directly from the index structure.

Currying and the Tensor Product

Embedding into the First Tensor Factor

Under the universal property of the triple tensor product, the map $v \mapsto v \otimes w \otimes x$ is linear precisely because the first argument slot of the corresponding trilinear map is linear; this is the mechanism by which the first slot of a trilinear map is identified with the leftmost tensor factor in $V \otimes W \otimes X$ .

Reduction to a Bilinear Map

Fixing only the first slot, rather than freezing it, produces the complementary object: a bilinear map in the remaining two arguments, $T (v_{0}, \cdot, \cdot) : W \times X \to U$ for a fixed $v_{0}$ , illustrating that isolating the first slot is dual to isolating the pair formed by the remaining two slots.

Role in Contraction and Evaluation

Slot-Specific Contraction

When a rank-three tensor is contracted against a covector in its first index, the resulting rank-two object is obtained by evaluating the first argument slot at a specific input and leaving the remaining two slots free; because the first slot is guaranteed linear, this contraction is well defined and commutes with linear combinations of the contracting covector.

Ordering Convention

The label "first argument slot" fixes an ordering convention that matters because, unlike a symmetric trilinear map, a general trilinear map need not treat its three slots interchangeably; distinguishing the first slot from the second and third is therefore essential whenever the trilinear map lacks full symmetry, which is the generic case for tensors of rank three.