9.21.3 Tensor Basis Index Notation

Tensor Basis Index Notation systematically represents tensor components with indices, enabling precise algebraic manipulation in multilinear contexts.

Tensor Basis Index Notation is the convention of attaching an index to each basis vector and each dual basis covector so that a tensor's expansion in a chosen basis can be written compactly, with the index serving both to label which basis element is meant and to pair with the matching index on the tensor's components.

Indexing the Basis Set

Ordering the Basis

A basis ({e_1, e_2, \dots, e_n}) is an ordered set: the index attached to each vector, (1) through (n), fixes a definite order in which the basis vectors are listed. This ordering is not intrinsic to the vector space itself but is a choice made when the basis is specified, and it must be kept consistent throughout any calculation.

{e_{i}}, i = 1, 2, \dots, n

Indexing the Dual Basis

Each dual basis covector (e^i) is indexed to match the basis vector (e_i) it is biorthogonal to, so that the index on the dual basis element directly identifies its partner in the original basis.

e^{i} (e_{j}) = δ_{j}^{i}

Index Position as a Grammatical Marker

Lower Index on Basis Vectors

The basis vector index is always written low, (e_i), which signals that basis vectors transform the same way as covariant quantities under a change of basis: directly with the transformation matrix rather than its inverse.

Upper Index on Dual Basis Vectors

The dual basis index is always written high, (e^i), signaling that dual basis covectors transform contravariantly, with the inverse transformation matrix, so that the biorthogonality relation (e^i(e_j) = \delta^i_j) is preserved under any change of basis.

e^{i}' = {(A^{- 1})}_{j}^{i} e^{j}

Index Matching in Tensor Expansions

Pairing Basis Index with Component Index

When a tensor is expanded in a basis, the basis index notation is what enables the Einstein summation convention: a repeated index, once as a subscript on the basis and once as a superscript on the component, indicates a sum over all values of that index.

v = v^{i} e_{i} = \sum_{i = 1}^{n} v^{i} e_{i}

Multiple Basis Indices for Higher-Rank Tensors

For a tensor built from tensor products of basis elements, each factor in the product carries its own index, and the tensor's full index notation lists all of them in the same order as the corresponding tensor product.

T = T_{j}^{i} e_{i} \otimes e^{j}

The index (i) on (e_i) matches the superscript (i) on (T^i_{\ j}), and the index (j) on (e^j) matches the subscript (j) on (T^i_{\ j}); both are summed, so this single expression stands for the full double sum over (i) and (j).

Visual Illustration

Significance for Consistency

Tensor basis index notation is the mechanism that keeps track of exactly which basis vector or dual basis covector each numerical component belongs to. Because the index position (upper for dual basis, lower for basis) and index letter must match precisely between basis elements and components for a summation to be valid, this notation acts as a built-in consistency check: an expression with mismatched or unmatched indices signals either a typo or an operation that is not tensorially well-defined.