9.3 Tensor Basis System Structure

Tensor Basis System Structure is the complete internal organization of a basis choice for tensor work: the primal basis of the vector space itself, the uniquely determined dual basis of the corresponding dual space, the family of induced tensor product bases built from these two for tensors of every order, and the single consistent transformation rule that propagates a change in the primal basis through all of these derived pieces simultaneously. It describes not any one basis in isolation but the whole interlocking apparatus that a single choice of basis for V automatically brings into existence for every tensor type built from V.

The Primal Basis as the Root of the Structure

One Choice Determines Everything Downstream

Selecting a basis {e₁, ..., eₙ} of the vector space V is the single free choice in the entire structure; every other component of the basis system — the dual basis, the induced tensor product bases, the specific numerical form of the metric — is either uniquely determined by this one choice or expressed relative to it, so that the basis system structure as a whole has exactly one genuine degree of freedom.

The Dual Basis Is Fixed, Not Chosen Independently

Given the primal basis, the dual basis {eⁱ} is not a second independent choice but the unique family of covectors satisfying eⁱ(eⱼ) = δⁱⱼ; the basis system structure therefore has no room for selecting a dual basis at odds with the primal one, and any apparent freedom in choosing covector components is really freedom already exhausted by the original choice of primal basis.

Induced Tensor Product Bases

One Basis for Every Tensor Type at Once

From the single pair {eᵢ} and {eⁱ}, an induced basis is automatically available for tensors of every type (p,q) simultaneously: the family {eᵢ₁ ⊗ ⋯ ⊗ eᵢₚ ⊗ eʲ¹ ⊗ ⋯ ⊗ eʲq}, indexed by all combinations of p upper and q lower basis labels, forms a basis of the corresponding tensor space, so a single choice of primal basis furnishes, without any further selection, a basis for scalars, vectors, covectors, and tensors of arbitrarily high order all at once.

Consistency Across Tensor Types

Because every induced basis in this family is built from the same underlying {eᵢ} and {eⁱ}, tensor operations that combine objects of different type — tensor products, contractions — automatically remain consistent across the whole structure: contracting an upper index of one tensor against a lower index of another, both expressed in their respective induced bases from the same primal choice, produces a result correctly expressed in the induced basis of the resulting lower-order tensor type, with no additional bookkeeping needed to reconcile the different bases involved.

How the Whole Structure Responds to a Change of the Primal Basis

A Single Jacobian Propagates Through Every Layer

Changing the primal basis by a Jacobian J induces a corresponding change in the dual basis by J⁻¹, and correspondingly determined changes in every induced tensor product basis, built from the appropriate combination of J and J⁻¹ factors matching the tensor type in question; the entire basis system structure transforms as a single coordinated unit under any change to its one genuine degree of freedom, the primal basis.

No Independent Transformation Rule for Derived Pieces

Because the dual and induced bases are uniquely determined by the primal basis rather than chosen separately, there is no separate transformation rule to derive for them; their transformation behavior follows automatically and mechanically from the primal basis's own transformation, which is why the tensor transformation law for a general (p,q) tensor is fully determined once the transformation rule for vectors (upper index) and covectors (lower index) is known.

Diagram of the Structure Radiating From One Choice

Why the Whole Structure Is Treated as a Single Unit

Consistency Is What Makes Index Notation Reliable

Index notation's rules — that a repeated upper-lower pair may be contracted, that a fully contracted expression is basis-independent, that components transform by the standard rule — all rely on the dual and induced bases being exactly the ones determined by the primal basis according to this fixed structure; substituting an ad hoc, independently chosen covector basis in place of the true dual basis would break these guarantees, which is precisely why the basis system structure treats the entire family of bases as a single, internally consistent unit generated from one initial choice rather than as several independently adjustable pieces.