2.13.3 Tensor Scalar Multiplication Field Dependence

Tensor Scalar Multiplication Field Dependence is the fact that the behavior and available scalars of the scalar multiplication operation are entirely determined by the choice of underlying field over which the vector space is defined, so that changing the field changes which scalings are possible and how they combine, even while the vector space's set of vectors and its addition operation remain unaffected. Field dependence highlights that a vector space is never considered in isolation from the field it is built over.

Formal Statement

Scalars Drawn Exclusively From the Field

The scalars available for scalar multiplication are precisely the elements of the field over which the vector space is defined, and no element outside that field can serve as a valid scalar for this operation.

Field Arithmetic Governs Coefficient Combination

Combining scalars, whether by adding them before scaling or multiplying them together before applying to a vector, is carried out entirely using the arithmetic operations of the underlying field.

Consequences of Changing the Field

Different Fields Yield Different Notions of Scalar

A vector space defined over one field cannot use scalars from an unrelated field directly, since scalar multiplication is only defined for elements belonging to the specific field fixed as part of the vector space's definition.

Impact on Dimension and Structure

The same underlying set of vectors, when regarded as a vector space over a different but related field, may exhibit a different dimension or structural behavior, since the available linear combinations depend on which field's scalars are permitted.

Field Properties That Support Scalar Multiplication

Reliance on Field Axioms

The distributive and associative laws required of scalar multiplication rely on the underlying field itself satisfying the axioms of a field, including the existence of multiplicative inverses for nonzero elements, which in turn supports operations such as solving for coefficients in linear combinations.

Characteristic of the Field and Its Effects

Properties of the field, such as its characteristic, can influence subtler aspects of vector space behavior, since certain combinations or cancellations available in one field, such as one with characteristic zero, may not be available in a field with different characteristic.

Role in Tensor Construction

Consistent Field Requirement Across Tensor Factors

When constructing a tensor from multiple vector spaces, all contributing spaces must share the same underlying field, since the multilinear operations defining the tensor product require scalar multiplication behavior to be compatible across every factor.

Field-Dependent Interpretation of Tensor Coefficients

The coefficients appearing in a tensor's coordinate representation are themselves elements of the shared underlying field, so the field dependence of scalar multiplication in each factor space directly determines the type of numbers that can appear as tensor components.

Summary of Key Properties

Scalars Anchored to a Specific Field

Tensor Scalar Multiplication Field Dependence establishes that the scalars usable in scalar multiplication, and the arithmetic governing their combination, are fixed entirely by the chosen underlying field of the vector space.

Prerequisite for Compatible Tensor Construction

This field dependence is why every vector space contributing to a tensor construction must be defined over the same field, ensuring that scalar multiplication behaves consistently throughout the entire construction.