4.6 Tensor Slotwise Linearity Property

Tensor Slotwise Linearity Property is the axiom, foundational to the entire theory of multilinear maps, stating that a tensor's defining map must satisfy additivity and homogeneity separately in each of its argument positions, with all other arguments held constant. It is the single property that qualifies a function of several vector arguments as a tensor in the first place, and every other tensor-theoretic notion, from components to contraction to symmetry, is built on top of this one requirement.

Formal Statement

Additivity in Each Slot

For a map $T : V_1 \times \cdots \times V_k \to F$, the property requires that for every slot index $i$ and every pair of vectors $u, w \in V_i$,

with all arguments in the omitted positions held fixed and identical on both sides.

Homogeneity in Each Slot

Simultaneously, for every scalar $c \in F$ and vector $u \in V_i$,

Together, additivity and homogeneity in each slot are the two halves of the ordinary linearity condition, applied position by position rather than to the tuple as a whole.

Why This Property Defines Tensors

The Universal Property Connection

The tensor product space $V_1 \otimes \cdots \otimes V_k$ is constructed precisely as the universal recipient of maps satisfying the slotwise linearity property: any function on the domain product satisfying this axiom factors uniquely through the tensor product. Without slotwise linearity, there would be no tensor product to factor through, and the function would simply be an arbitrary, unstructured mapping of several variables.

Distinguishing Tensors from General Functions

A generic scalar-valued function of $k$ vector arguments need satisfy no algebraic constraint at all. The slotwise linearity property is what selects, from among all such functions, the very small and highly structured subclass that qualifies as tensors: this subclass forms a vector space in its own right, of finite dimension equal to the product of the dimensions of the $V_i$, in sharp contrast to the space of arbitrary functions, which is typically infinite-dimensional.

Structural Consequences

Reduction to Finite Component Data

Because the property holds in every slot, expanding each argument in a chosen basis and applying additivity and homogeneity repeatedly reduces the evaluation of $T$ on any tuple to a finite sum over the tensor's components, each weighted by a product of basis coordinates. Slotwise linearity is exactly the axiom that guarantees this reduction is always possible, no matter how the arguments are chosen.

Vanishing at Zero and Sign Behavior

Slotwise linearity forces $T$ to output zero whenever any single argument is the zero vector, and it forces $T(\ldots, -u, \ldots) = -T(\ldots, u, \ldots)$ in every slot, since these are direct consequences of linearity applied to that one position. These vanishing and sign properties hold regardless of what values occupy the other slots.

Compatibility With Linear Transformations

If $L_i : V_i \to V_i'$ is a linear map for each slot, precomposing $T$ with the tuple of maps $(L_1, \ldots, L_k)$ produces a new multilinear map on $V_1' \times \cdots \times V_k'$ pulled back through each $L_i$. Slotwise linearity is exactly the property that makes this pullback construction well-defined and itself slotwise linear, which underlies how tensors transform consistently under change of basis or change of coordinate system.

Summary of Key Points

• The slotwise linearity property requires additivity and homogeneity in each argument slot separately, with all other slots fixed.
• It is the defining axiom that qualifies a multi-argument function as a tensor, distinguishing tensors from the vastly larger space of unconstrained functions.
• It underlies the universal property connecting multilinear maps to the tensor product space.
• It guarantees that every tensor can be reduced to a finite table of components once a basis is fixed.
• It ensures tensors vanish on zero arguments and transform predictably under precomposition with linear maps in each slot.