1.4.1 Coordinate Independent Tensor Description

Coordinate Independent Tensor Description offers a framework to describe tensors without coordinates, focusing on their intrinsic geometric and algebraic properties.

Coordinate Independent Tensor Description is the formulation of a tensor as an abstract mathematical object, either a multilinear map on copies of a vector space and its dual, or an element of the corresponding tensor product space, defined entirely without reference to any basis, coordinate system, or numerical array. This description treats the tensor itself as the primary object of study, with any coordinate representation regarded as a secondary, chosen, and ultimately dispensable way of exhibiting the tensor's numerical content relative to an arbitrary frame of reference.

The Two Coordinate-Free Formulations

As a Multilinear Map

A tensor of type (p, q) can be described coordinate-independently as a multilinear map

T : \times_{i = 1}^{p} V^{*} \times \times_{j = 1}^{q} V \to F

This formulation defines T purely by its action on arbitrary vector and covector arguments, with no coordinate system entering the definition at any stage; only when specific arguments are chosen and evaluated does a number emerge.

As an Element of a Tensor Product Space

Equivalently, a tensor can be described coordinate-independently as a single element of the abstractly constructed vector space T^p_q(V), built as the tensor product of p copies of V and q copies of V*. This space is defined by a universal property relating multilinear maps out of the product of vector spaces to linear maps out of the tensor product, a characterization that makes no mention of coordinates.

T \in T_{q}^{p} (V)

Why the Description Is Coordinate Independent

No Basis Appears in the Definition

Neither the multilinear map characterization nor the universal property defining the tensor product space refers to a basis of V at any point; both formulations are stated purely in terms of the vector space structure of V and V* themselves, which exist independently of any particular coordinate system chosen to describe them.

Basis-Dependence Enters Only Upon Evaluation

A coordinate system enters the picture only when someone chooses to evaluate the multilinear map on specific basis vectors, or to expand the tensor product element in terms of a specific basis, producing the array of components. The tensor itself, prior to this choice, remains a single well-defined object.

Basis Independence of Tensor Operations

Addition and Scalar Multiplication

The sum of two tensors and the scalar multiple of a tensor are defined directly at the level of the abstract objects, requiring no coordinate system: for multilinear maps, the sum is the map sending each argument tuple to the sum of the two individual outputs, and the scalar multiple is the map sending each argument tuple to the scaled output.

(S + T) (\dots) = S (\dots) + T (\dots)

Tensor Product and Contraction

The tensor product of two tensors is defined directly as the multilinear map that separately evaluates each factor on its own set of arguments and multiplies the results, and contraction is defined directly via the natural pairing between V and V*, both operations requiring no coordinate system to be stated or carried out.

Contrast with Coordinate Description

Two Views, One Object

The coordinate-independent description and the coordinate description via components are not competing definitions of a tensor but two views of the same object: the coordinate description is obtained from the coordinate-independent one by evaluating the multilinear map on a chosen basis, and the coordinate-independent object can always be reconstructed from any one of its coordinate descriptions together with the basis used to produce it.

Guaranteeing Consistency Across Bases

Because the coordinate-independent description exists prior to any basis choice, two different coordinate descriptions of the same tensor, computed in two different bases, are guaranteed to be related by the standard transformation law; this guarantee is precisely what the coordinate-independent formulation is designed to encode and certify.

Practical Value of the Coordinate-Free Viewpoint

Stating Universal Identities

Identities and relations among tensors that are meant to hold regardless of coordinate choice are most naturally stated and proved using the coordinate-independent description, since no argument about how the identity behaves under a change of basis is needed; the identity is simply true of the abstract objects involved.

Clarifying Geometric Meaning

The coordinate-independent description emphasizes the geometric or structural role a tensor plays, such as measuring lengths, encoding a linear transformation, or defining a volume, separating this intrinsic meaning from the incidental numerical values that appear once an arbitrary coordinate system is imposed.

Diagrammatic Summary

The diagram shows a single coordinate-independent tensor T giving rise to two different arrays of components when evaluated relative to two different bases, illustrating that the abstract object remains unchanged while its coordinate representation varies.