4.4.3 Tensor Higher Arity Slotwise Linearity

Tensor Higher Arity Slotwise Linearity is the defining condition that a higher-arity multilinear map must satisfy: linearity holds in each individual argument slot when every other slot is held fixed, without requiring or implying linearity in the arguments taken jointly. It is the precise technical meaning of the word "multilinear," and it is the property that separates a genuine higher-arity tensor from an arbitrary function of several vector variables.

Formal Definition

The Slotwise Condition

Let $T : V_1 \times \cdots \times V_k \to F$ be a map of arity $k$. Slotwise linearity requires that, for every slot index $i \in {1, \ldots, k}$, and for all fixed vectors $v_1, \ldots, v_{i-1}, v_{i+1}, \ldots, v_k$, the induced single-variable map

is a linear map $V_i \to F$. Equivalently, additivity and homogeneity must hold in each slot separately:

for every scalar $c \in F$, applied independently to each of the $k$ positions.

Why "Slotwise" and Not "Jointly Linear"

A map that were linear on the domain product treated as a single vector space (i.e., linear jointly in all arguments at once) would have to satisfy $T(v_1 + w_1, \ldots, v_k + w_k) = T(v_1, \ldots, v_k) + T(w_1, \ldots, w_k)$, which is a much stronger and generally false condition for $k \geq 2$. Slotwise linearity deliberately stops short of this: it only requires linearity when exactly one argument varies and the rest are frozen.

Consequences of Slotwise Linearity

Vanishing on Zero in Any Slot

Because each slot map is linear, and linear maps send $0$ to $0$, a direct consequence is that $T$ vanishes whenever any single argument is the zero vector, regardless of the values in the remaining slots:

Multilinear Expansion Over a Basis

Slotwise linearity is exactly the property that permits full expansion of $T$ over a chosen basis: each argument can be expanded in coordinates, and the linearity in each slot separately allows the coordinates to be pulled out one slot at a time, producing the full multi-index sum over basis evaluations described elsewhere for the domain product of a higher-arity map. Without slotwise linearity in every slot, no such reduction to a finite table of components would be possible.

Non-Linearity of the Overall Map

Slotwise linearity does not make $T$ linear as a function of the whole tuple. For $k \geq 2$, scaling all arguments simultaneously by $c$ scales the output by $c^{k}$, not by $c$:

This homogeneity of degree $k$ in the joint arguments is a hallmark that distinguishes true multilinear maps from ordinary linear maps, and it grows more pronounced as arity increases.

Relation to Higher Arity Structure

Building Blocks of Symmetric and Antisymmetric Maps

Symmetric and antisymmetric higher-arity tensors are defined by imposing permutation conditions on top of an already slotwise-linear map; slotwise linearity is the prerequisite structure without which symmetry properties could not even be meaningfully stated, since symmetry compares the map's behavior across slots that must already individually be linear.

Role in Contraction and Composition

Operations such as contraction, tensor product, and composition with linear maps in individual slots all rely on slotwise linearity to be well defined: substituting a linear combination of basis vectors into one slot, applying a linear map to one argument before evaluating $T$, or summing over one paired index, are all manipulations that are only guaranteed to behave predictably because linearity holds separately in that one position.

Summary of Key Points

• Slotwise linearity means $T$ is linear in each argument individually, with all other arguments fixed.
• It is strictly weaker than joint linearity over the whole domain product, and it is precisely what "multilinear" means.
• It forces $T$ to vanish whenever any single slot is the zero vector.
• It permits full expansion of $T$ into a finite table of components once a basis is fixed.
• Scaling every argument by $c$ scales the output by $c^{k}$, reflecting the arity-dependent homogeneity of the map.