4.15.2 Tensor Basis Value Assignment

Tensor Basis Value Assignment defines how tensors are expressed through basis vectors, establishing their components in a structured algebraic framework.

Tensor Basis Value Assignment is the specific data of a scalar assigned to every tensor basis input tuple, considered as a free choice that can be made arbitrarily, without constraint, and that uniquely determines a tensor once made. It is the concrete object underlying tensor multilinear basis determination: rather than describing the general principle that basis values determine a tensor, it describes the assignment itself, emphasizing that any conceivable assignment of numbers to basis input tuples is permissible and gives rise to a genuine multilinear map.

Defining a Basis Value Assignment

The Assignment as a Function on Index Combinations

For a type (p, q) tensor on an n-dimensional vector space V with basis e_1, ..., e_n, a basis value assignment is a function

A : (1, \dots, n)_{}^{p + q times} \to F

sending each combination of indices (i_1, ..., i_p, j_1, ..., j_q) to a scalar A(i_1, ..., i_p, j_1, ..., j_q) in the base field F. No relationship is required between the values assigned to different index combinations; the assignment is simply a table of n^{p+q} numbers, one for each basis input tuple.

Complete Freedom in the Choice of Values

Every entry of the assignment can be chosen entirely independently of every other entry: there is no symmetry, antisymmetry, sum-to-zero, or any other constraint imposed on a basis value assignment in general, and any function A of this form, no matter how arbitrarily its values are chosen, constitutes a valid basis value assignment.

From Assignment to Tensor

Defining a Tensor from the Assignment

Given a basis value assignment A, a tensor T is defined on arbitrary arguments by declaring the component evaluation formula, with A in place of the component array, to be the value of T:

T (α^{1}, \dots, v_{q}) = \sum_{i_{1}, \dots, j_{q}}^{⋮} A (i_{1}, \dots, j_{q}) α_{1}^{i_{1}} \dots v_{q}^{j_{q}}

with the coordinates of the arguments substituted as usual. This definition uses A directly, without modification, as the collection of component values that the resulting tensor is required to have.

Automatic Recovery of the Assignment as Basis Values

Evaluating the resulting T on the basis input tuple corresponding to indices (i_1, ..., i_p, j_1, ..., j_q) returns exactly A(i_1, ..., i_p, j_1, ..., j_q), since every other term in the component evaluation formula vanishes: any coordinate of a basis vector or basis covector relative to its own basis is 1 in the matching position and 0 elsewhere, by the definition of the Kronecker delta pairing, leaving only the single term matching the chosen basis input tuple.

Well-Definedness of the Constructed Tensor

Multilinearity Follows Automatically

The map T constructed from a basis value assignment A is automatically multilinear, since the component evaluation formula defining it is linear, to the first power, in the coordinates of each argument separately; this guarantees both the tensor multilinear additivity property and the tensor multilinear homogeneity property hold in every slot, regardless of what values A happens to assign.

No Assignment Fails to Produce a Tensor

Because multilinearity of the constructed map does not depend on any particular property of A, every basis value assignment, however arbitrary, produces a well-defined tensor; there is no assignment of numbers to basis input tuples that fails to correspond to some tensor, which is precisely the existence half of basis determination.

Assignment and Tensor as Interchangeable Data

A Bijective Correspondence

The process of extracting a basis value assignment from a tensor by evaluating it on every basis input tuple, and the process of constructing a tensor from a basis value assignment via the component evaluation formula, are mutually inverse; a tensor and its basis value assignment carry exactly the same information, and either one can be recovered completely from the other.

Practical Identification in Computation

In practical computation, a tensor is frequently represented directly by its basis value assignment, stored as a multidimensional array of numbers, since this assignment is both necessary and sufficient to reconstruct every property and every evaluation of the tensor, making the assignment itself the natural computational stand-in for the abstract tensor.

Diagrammatic Summary

The diagram shows an arbitrary basis value assignment on the left constructing exactly one tensor on the right, illustrating that no restriction on the chosen values is needed for the construction to succeed.