3.1.4 Tensor Dual Space Scalar Action Structure

Tensor Dual Space Scalar Action Structure is the description of how scalar multiplication acts on a dual tensor space such as V* ⊗ W*, covering the freedom to move a scalar between the two factors of a simple element without changing the element itself, the resulting scalar-multiplication rule on the coordinates of a general element, and the compatibility of this scalar action with the interpretation of dual tensor space elements as multilinear functionals, where scaling the tensor scales the functional's output by the same factor. This structure is the multiplicative counterpart to additive structure on the dual tensor space, and together the two give the dual tensor space its full vector-space character.

Scalar Absorption Between Factors

The Basic Identity

For a simple element ω ⊗ η ∈ V* ⊗ W* and a scalar c ∈ F, bilinearity of the tensor product gives:

so the same overall element results whether the scalar is applied to ω, to η, or left outside the tensor product entirely. This freedom means a given simple element of V* ⊗ W* has infinitely many different decompositions as (c ω) ⊗ (η / c) for nonzero c, all denoting the identical element.

Consequence for Simple-Element Identification

Because of this absorption freedom, the pair (ω, η) is not uniquely determined by the simple element ω ⊗ η; only the element itself is well defined, and any two pairs related by (ω', η') = (cω, η/c) for nonzero c produce the same simple element. Care in distinguishing the element ω ⊗ η from the pair (ω, η) that produced it is a standing requirement whenever scalar action is discussed for simple elements.

Scalar Multiplication in Coordinates

Entrywise Scaling

For a general element β = Σ_{i,j} b_{ij} e^i ⊗ f^j ∈ V* ⊗ W*, expressed in the dual basis induced from bases {e_i} of V and {f_j} of W, scalar multiplication acts entrywise on the coordinates:

matching the coordinatewise scalar multiplication used for any vector space once a basis is fixed. This is a direct consequence of scalar multiplication being distributed across the terms of the coordinate sum, together with the entry-by-entry absorption rule for each individual simple term.

Matrix Description

When β corresponds to a matrix (b_{ij}) under the coordinate identification described in tensor dual space element structure, cβ corresponds to the scalar matrix multiple (c b_{ij}); scalar multiplication of a dual tensor space element is, in coordinates, exactly the familiar operation of multiplying every entry of a matrix by a scalar.

Compatibility With the Functional Interpretation

Scaling the Output of a Multilinear Functional

Regarding β ∈ V* ⊗ W* as a bilinear functional on V × W, scalar action satisfies:

so scaling the tensor by c scales every output value of the corresponding functional by the same c, uniformly across all input pairs (v, w). This matches the ordinary meaning of a scalar multiple of a function, confirming that the functional interpretation and the tensor-space interpretation of scalar action agree exactly.

Zero Scalar and the Zero Functional

Multiplying any element β by the scalar 0 produces the zero element of V* ⊗ W*, which corresponds to the identically-zero functional, (0, v, w) ↦ 0 for every input pair; this is consistent both with the general vector-space axiom 0 · x = 0 and with the direct computation (0β)(v, w) = 0 · β(v, w) = 0.

Interaction With Rank Under Scalar Action

Rank Is Unchanged by Nonzero Scaling

For nonzero c, the rank of cβ, the minimal number of simple elements needed in a decomposition, equals the rank of β, since scaling any minimal decomposition β = Σ ω_k ⊗ η_k by c gives cβ = Σ (c ω_k) ⊗ η_k, still a sum of the same number of simple elements, and no shorter decomposition of cβ could exist without producing, after scaling by 1/c, a shorter decomposition of β itself. Only multiplication by the scalar 0 collapses rank, sending every element to the rank-0 zero element regardless of its original rank.