9.21 Tensor Basis Coordinate Notation

Tensor Basis Coordinate Notation is the systematic convention for writing the components of a tensor relative to a chosen basis of the underlying vector space, together with the corresponding dual basis of the covector space. It specifies how upper (contravariant) and lower (covariant) indices attach to basis vectors and basis covectors respectively, and how the full tensor is reconstructed as a linear combination of basis tensor products weighted by its components.

Basis Vectors and Dual Basis Covectors

Basis of the Vector Space

Given an (n)-dimensional vector space (V), a basis is an ordered set of linearly independent vectors ({e_1, e_2, \dots, e_n}) that spans (V). Any vector (v \in V) can be written uniquely as a linear combination of these basis vectors.

The scalars (v^i) are the contravariant components of (v), and they carry an upper index because they transform inversely to the basis vectors under a change of basis.

Dual Basis Covectors

The dual space (V^) consists of linear functionals on (V). Associated with the basis ({e_i}) is a unique dual basis ({e^1, e^2, \dots, e^n}) of (V^), defined by the biorthogonality condition.

where (\delta^i_j) is the Kronecker delta, equal to (1) when (i = j) and (0) otherwise. A covector (\omega \in V^*) is expressed in terms of this dual basis with lower-indexed components (\omega_i).

Component Notation for General Tensors

Mixed Tensor Components

A tensor of type ((p, q)), with (p) contravariant slots and (q) covariant slots, is expressed as a sum over tensor products of (p) basis vectors and (q) dual basis covectors, weighted by its components (T^{i_1 \dots i_p}_{\ j_1 \dots j_q}).

The index placement is not cosmetic: upper indices always pair with basis vectors (e_i), and lower indices always pair with dual basis covectors (e^j). This pairing is what allows the abstract tensor (T) to be reconstructed unambiguously from its component array once a basis is fixed.

Einstein Summation Convention

Because sums of this form appear constantly, tensor basis coordinate notation is almost always combined with the Einstein summation convention: whenever an index appears once as a superscript and once as a subscript within the same term, summation over that index is implied and the summation symbol is omitted.

This convention keeps expressions compact while preserving full information about which index ranges are being contracted.

Coordinate Notation Under a Change of Basis

Transformation of Basis Vectors

Suppose a new basis ({e'i}) is related to the old basis ({e_i}) by an invertible transformation matrix (A), with entries (A^j{\ i}).

Transformation of Components

For the vector (v) to remain the same geometric object regardless of the basis used to describe it, its contravariant components must transform with the inverse matrix (A^{-1}), whose entries are written ((A^{-1})^i_{\ j}).

This inverse relationship is the defining property of a contravariant index: components change oppositely to the basis so that the sum (v^i e_i) is invariant. Covariant components, such as those of a covector or the lower indices of a mixed tensor, transform directly with (A) rather than its inverse, which is why they are called covariant.

Diagrammatic View of Basis Coordinates

A vector expressed in a two-dimensional basis can be visualized as a linear combination of the two basis directions, with the coefficients being exactly the contravariant components in that basis.

In the diagram, the dashed vector (v) is reconstructed by scaling (e_1) by the component (v^1) and (e_2) by the component (v^2), then adding the results. Changing the basis vectors (e_1) and (e_2) changes the numeric values of (v^1) and (v^2) needed to describe the same fixed arrow (v).

Why the Notation Matters

Basis Independence of the Underlying Tensor

Tensor basis coordinate notation makes explicit that the array of numbers (T^{i_1 \dots i_p}_{\ j_1 \dots j_q}) is not the tensor itself but only its representation in a particular basis. The tensor (T) is the full sum including the basis vectors and covectors; the components alone are basis-dependent data that only acquire meaning once paired with the basis they were computed in.

Consistency Across Operations

Because upper and lower indices transform in inverse ways, contractions such as (T^i_{\ i}) (summing an upper index against a matching lower index) produce basis-independent scalars. This consistency is what permits operations like the trace, raising and lowering of indices via a metric tensor, and tensor contraction to be defined purely in terms of index manipulation, without reference to any particular coordinate system.