2.13.5 Tensor Scalar Multiplication Tensor Input Role

Tensor Scalar Multiplication Tensor Input Role is the function scalar multiplication performs when it is applied to vectors that will serve as inputs to a multilinear map or tensor construction, requiring that scaling one input by a field element before applying the map produces the same result as applying the map first and then scaling the output by that same element. This homogeneity-in-each-input requirement is the second half of the multilinearity condition, complementing additivity in each input.

Formal Statement

Homogeneity in a Single Input Slot

For a multilinear map with several vector arguments, scaling one argument by a scalar before applying the map produces an output equal to the output obtained without scaling, multiplied afterward by that same scalar, while holding all other arguments fixed.

Applies Independently to Each Slot

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

Relationship to Ordinary Linearity

Generalization From Single-Variable Linear Maps

Homogeneity of this kind generalizes the scalar compatibility 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 Additivity for Full Multilinearity

Homogeneity in each input, together with the corresponding additivity condition under vector addition 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

Extracting Scalars Through the Tensor Product

Because the tensor product operation is itself multilinear, a scalar multiplying one factor can be pulled out to multiply the entire elementary tensor, and the position from which the scalar is extracted does not affect the final result.

Enabling Coefficient Extraction When Expanding Tensors in a Basis

This homogeneity across factors is what allows the scalar coefficients from each vector's basis expansion to be gathered together into a single combined coefficient when an elementary tensor is rewritten in terms of a tensor product basis, simplifying the coordinate description of the tensor.

Role in Broader Vector Space Structure

Consistency With General Scalar Action Laws

This homogeneity-in-each-input role is fully consistent with, and indeed derived from, the scalar action laws already governing ordinary scalar multiplication, 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 homogeneity in each input, alongside additivity, 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

Homogeneity as Half of Multilinearity

Tensor Scalar Multiplication Tensor Input Role captures the scalar-compatible half of the multilinearity condition that governs how scaling in any single input interacts with tensor-producing maps.

Enabling Coefficient Extraction in Tensor Expressions

This homogeneity is what ultimately permits scalar coefficients to be freely moved between individual tensor factors and the tensor as a whole, simplifying computation throughout tensor algebra.