4.15.1 Tensor Basis Input Tuple

A Tensor Basis Input Tuple defines tensor components in a specific basis through indexed input.

Tensor Basis Input Tuple is a tensor multilinear input tuple in which every entry is drawn directly from a fixed basis of the underlying vector space or its dual, rather than being an arbitrary vector or covector, so that each slot of the tuple is filled by exactly one basis vector or one basis covector. It is the special class of input tuples whose associated evaluation results constitute the basis values referenced in tensor multilinear basis determination, making it the essential vehicle through which a tensor's component description is obtained.

Structure of a Basis Input Tuple

Composition for a Type (p, q) Tensor

For a type (p, q) tensor T on a vector space V with basis e_1, ..., e_n and dual basis e^1, ..., e^n, a basis input tuple selects, for each of the p contravariant slots, one basis covector e^{i_r}, and for each of the q covariant slots, one basis vector e_{j_s}:

(e^{i_{1}}, \dots, e^{i_{p}}, e_{j_{1}}, \dots, e_{j_{q}})

with each index i_r and j_s independently ranging from 1 to n. Unlike a general input tuple, every entry here is one of the finitely many basis elements, never an arbitrary linear combination of them.

Total Number of Distinct Basis Input Tuples

Because each of the p + q slots independently ranges over n possible basis choices, the total number of distinct basis input tuples for a type (p, q) tensor on an n-dimensional space is n^{p+q}, matching exactly the number of components in the tensor's component array and the dimension of the tensor product space containing T.

Basis Input Tuples and the Basis Value Table

Evaluating on a Basis Input Tuple

Applying the tensor multilinear evaluation operation to T on a basis input tuple produces exactly one entry of the basis value table used in tensor multilinear basis determination:

T (e^{i_{1}}, \dots, e_{j_{q}}) = T_{j_{1} \dots j_{q}}^{i_{1} \dots i_{p}}

so that the entire component array of T is nothing more than the collection of evaluation results obtained by feeding every possible basis input tuple into T in turn.

Basis Input Tuples as the Building Blocks of Determination

Since every general input tuple can be expanded, through additivity and homogeneity, as a linear combination of basis input tuples, the values of T on basis input tuples alone are sufficient to reconstruct the value of T on any general input tuple, which is precisely the content of tensor multilinear basis determination applied at the level of tuples rather than of tensors.

Basis Input Tuples Versus General Input Tuples

Special Case with No Coordinate Ambiguity

A general input tuple involves a coordinate for every basis direction in every slot, requiring a full set of coordinates to specify each argument, whereas a basis input tuple requires only a single index per slot, since each entry is exactly one basis element rather than a combination; a basis input tuple can therefore be specified completely by a tuple of indices, (i_1, ..., i_p, j_1, ..., j_q), rather than by a tuple of coordinate arrays.

Recovering a General Input Tuple's Result from Basis Input Tuples

The evaluation of T on a general input tuple decomposes, via the component evaluation formula, into a sum over basis input tuples, each weighted by a product of coordinates drawn from the general tuple's entries; the result on the general tuple is thus a weighted combination of the results on the finitely many basis input tuples, rather than an independent piece of information.

Enumerating Basis Input Tuples

Systematic Listing by Index Combinations

Because each slot's basis choice is independent of every other slot's choice, the full collection of basis input tuples can be enumerated systematically by iterating over every combination of indices, one per slot, in a fixed order, which is exactly the enumeration used to build up the component array of a tensor entry by entry.

Relation to Diagonal Input Tuples

A basis input tuple is a special case of an input tuple more generally, and it should be distinguished from a diagonal input tuple, which repeats the same vector or covector across several slots of the same kind; a basis input tuple may or may not be diagonal, depending on whether the same basis index happens to be chosen for more than one slot of the same kind.

Diagrammatic Summary

The diagram shows a basis input tuple as a sequence of slots, each filled with exactly one basis covector or basis vector, contrasting with a general input tuple whose entries would instead be arbitrary linear combinations of these basis elements.