2.12.5 Tensor Vector Addition Tensor Input Role

Tensor Vector Addition Tensor Input Role is the function vector addition performs when it is applied to vectors that will serve as inputs to a multilinear map or tensor construction, requiring that the map's behavior on a sum of two inputs in a given slot matches the sum of the map's outputs on each input taken separately in that slot. This additivity-in-each-input requirement is one half of the multilinearity condition that defines how tensors interact with the vector spaces feeding into them.

Formal Statement

Additivity in a Single Input Slot

For a multilinear map with several vector arguments, replacing one argument with a sum of two vectors produces an output equal to the sum of the outputs obtained by using each of the two vectors separately in that same slot, while holding all other arguments fixed.

Applies Independently to Each Slot

This additivity requirement is imposed separately on every input slot of a multilinear map, so a map with several vector arguments must satisfy the sum rule individually in each argument position, holding the remaining positions fixed each time.

Relationship to Ordinary Linearity

Generalization From Single-Variable Linear Maps

Additivity of this kind generalizes the additivity condition already familiar from ordinary linear maps of one variable, extending it to maps that take several vector arguments simultaneously, one from each of several vector spaces.

Combined With Homogeneity for Full Multilinearity

Additivity in each input, together with the corresponding homogeneity condition under scalar multiplication in each input, together constitute the full multilinearity that tensor-defining maps and the tensor product itself are required to satisfy.

Consequences for Tensor Construction

Distributing Sums Through the Tensor Product

Because the tensor product operation is itself multilinear, a sum of vectors appearing in one factor position distributes across the tensor product, so the tensor of a sum in one slot equals the sum of the tensors formed using each addend separately in that slot.

Enabling Expansion of Tensors in a Basis

This distributive behavior over addition in each factor is what allows any element of a tensor product space to be expanded as a sum of elementary tensors formed from basis vectors, since any vector can first be written as a sum of scaled basis vectors and this sum can then be distributed through the tensor product.

Role in Broader Vector Space Structure

Consistency With General Vector Addition Laws

This additivity-in-each-input role is fully consistent with, and indeed derived from, the algebraic laws already governing ordinary vector addition, applied here at the level of an entire multilinear or tensor-producing map rather than a single vector.

Prerequisite for Recognizing a Map as Genuinely Multilinear

Verifying additivity in each input is a required step when confirming that a given map qualifies as multilinear, and hence as eligible to define or interact meaningfully with a tensor construction.

Summary of Key Properties

Additivity as Half of Multilinearity

Tensor Vector Addition Tensor Input Role captures the additive half of the multilinearity condition that governs how vector addition in any single input interacts with tensor-producing maps.

Enabling Structural Expansion of Tensors

This additivity is what ultimately permits tensors to be expanded into sums of elementary tensors built from basis vectors, connecting vector addition directly to the practical construction of tensor components.