2.3.3 Tensor Vector Space Scalar Action

Tensor Vector Space Scalar Action describes how scalars interact with tensors, scaling their components within a structured vector space framework.

Tensor Vector Space Scalar Action is the operation by which an element of the underlying field acts on a tensor to rescale it, producing a new tensor of the same type whose components are each multiplied by that field element. It is the second of the two operations, alongside tensor addition, that endow the collection of tensors of a fixed type with the structure of a vector space, and it generalizes the familiar scalar multiplication of ordinary vectors to multilinear objects of arbitrary rank.

Formal Definition

Setting

Let $V$ be a finite-dimensional vector space over a field $F$ , and let $T_{s}^{r} (V)$ denote the space of tensors of type $(r, s)$ over $V$ , realized as multilinear maps

T : \underset{r factors}{V^{*} \times \dots \times V^{*}} \times \underset{s factors}{V \times \dots \times V} \to F

Definition of the Action

For a scalar $α \in F$ and a tensor $T \in T_{s}^{r} (V)$ , the scalar action produces the tensor $α T$ defined pointwise by

(α, T) (ω^{1}, \dots, ω^{r}, v_{1}, \dots, v_{s}) = α \cdot T (ω^{1}, \dots, v_{s})

Because scaling the output of a multilinear map by a fixed field element preserves multilinearity in every slot, $α T$ is again a well-defined tensor of the same type $(r, s)$ .

Componentwise Description

Component Formula

Relative to a basis $e_{1}, \dots, e_{n}$ of $V$ and its dual basis, a tensor is represented by its components $T_{j_{1} \dots j_{s}}^{i_{1} \dots i_{r}}$ . The scalar action acts by multiplying every component by the scalar:

{(α, T)}_{j_{1} \dots j_{s}}^{i_{1} \dots i_{r}} = α \cdot T_{j_{1} \dots j_{s}}^{i_{1} \dots i_{r}}

Basis Independence

Under a change of basis, the components of $T$ transform by a multilinear transformation law built from the change-of-basis matrix and its inverse. Since every component is scaled by the same factor $α$ before this law is applied, the transformation law commutes with the scalar action, so the resulting tensor $α T$ is independent of the basis chosen to compute it.

Special Cases

Vectors

When $(r, s) = (1, 0)$ , the tensor space is $V$ itself, and the scalar action reduces to the ordinary scalar multiplication of a vector by a field element.

Covectors

When $(r, s) = (0, 1)$ , the scalar action reduces to scaling a linear functional in the dual space $V^{*}$ .

Matrices and Bilinear Forms

When $(r, s) = (1, 1)$ , tensors correspond to linear endomorphisms of $V$ , represented in a basis as matrices, and the scalar action coincides with ordinary scalar multiplication of a matrix, in which every entry is multiplied by $α$ .

Algebraic Properties

Compatibility with Field Multiplication

For scalars $α, β \in F$ and a tensor $T$ , successive scalar actions compose according to multiplication in the field:

α (β, T) = (α, β) T

Distributivity over Tensor Addition

The scalar action distributes over the sum of two tensors of the same type:

α (S + T) = α S + α T

Distributivity over Scalar Addition

It likewise distributes over addition of scalars:

(α + β) T = α T + β T

Multiplicative Identity

The multiplicative identity of the field acts trivially on every tensor:

1 T = T

Absorbing Element

Scaling by the zero element of the field always yields the zero tensor, regardless of the tensor being scaled:

0 T = 0

Relation to Vector Space Structure

The Vector Space Axioms

Together with tensor addition, the scalar action satisfies all of the vector space axioms over $F$ : closure under both operations, associativity and commutativity of addition, existence of a zero tensor and additive inverses, compatibility of scalar multiplication with field multiplication, and the two distributive laws. These axioms are precisely what allow $T_{s}^{r} (V)$ to be treated as a vector space rather than merely a set of multilinear maps.

Scaling a Simple Tensor

For a simple, or decomposable, tensor formed from a tensor product of vectors and covectors, the scalar action may be absorbed into any single one of the constituent factors, since the tensor product is itself multilinear:

α (v_{1} \otimes v_{2}) = (α, v_{1}) \otimes v_{2} = v_{1} \otimes (α, v_{2})

Restriction to a Common Type

The scalar action always maps a tensor of type $(r, s)$ to another tensor of the same type; it never changes the number of vector or covector arguments a tensor accepts, since scaling only rescales the multilinear map's output values, not its domain structure.