1.4.4 Tensor Basis Description

Tensor Basis Description outlines the foundational structure of tensors, explaining how they are built from basis vectors and their role in multilinear algebra.

Tensor Basis Description is the specification of how a tensor is exhibited relative to a chosen basis of its underlying vector space by expressing it as an explicit linear combination of basis tensor products, formed from the basis vectors of V and the corresponding dual basis covectors of V*, weighted by the tensor's components. This description bridges the coordinate-free formulation of a tensor, as a multilinear map or an abstract element of a tensor product space, and its coordinate formulation, as a bare array of numbers, by making explicit the basis elements that the array of numbers is understood to multiply.

Constructing the Basis of T^p_q(V)

From a Basis of V to a Basis of the Tensor Space

Given a basis e_1, ..., e_n of V, the dual basis e^1, ..., e^n of V* is uniquely determined by requiring e^i(e_j) to equal 1 when i = j and 0 otherwise. From these two bases, a basis for the tensor space T^p_q(V) is built by forming every possible tensor product of p factors drawn from e_1, ..., e_n and q factors drawn from e^1, ..., e^n.

e_{i_{1}} \otimes \dots \otimes e_{i_{p}} \otimes e^{j_{1}} \otimes \dots \otimes e^{j_{q}}

for every combination of index values from 1 to n, producing exactly n^(p+q) distinct basis tensor products, matching the dimension of T^p_q(V).

The Full Basis Description of a Tensor

Any tensor T in T^p_q(V) is described relative to this basis by the unique expansion

T = T_{j_{1} \dots j_{q}}^{i^{1} \dots i^{p}} e_{i_{1}} \otimes \dots \otimes e_{i_{p}} \otimes e^{j_{1}} \otimes \dots \otimes e^{j_{q}}

under the Einstein summation convention, with the coefficients being precisely the components of T relative to this basis, and this expansion is what most fully constitutes the tensor's basis description.

Examples at Low Rank

Vector Basis Description

A type (1, 0) tensor v has basis description v = v^i e_i, expressing the vector as a sum of basis vectors weighted by its components.

Covector Basis Description

A type (0, 1) tensor ω has basis description ω = ω_i e^i, expressing the covector as a sum of dual basis covectors weighted by its components.

Rank-Two Basis Description

A type (1, 1) tensor T has basis description T = T^i_j e_i ⊗ e^j, summed over both i and j, expressing the tensor as a sum over all n^2 basis tensor products e_i ⊗ e^j, each weighted by the corresponding component.

Change of Basis Description

Relating Two Basis Descriptions

If a second basis f_1, ..., f_n of V is related to the first by f_k = A^i_k e_i, then the basis tensor products built from f_1, ..., f_n and its dual basis are related to those built from e_1, ..., e_n by the same change-of-basis matrix, and the basis description of a fixed tensor T in terms of the new basis tensor products uses new components computed by the standard transformation law, while the abstract sum T itself is unchanged.

T = T_{j_{1} \dots}^{i^{1} \dots} e_{i_{1}} \otimes \dots = {\tilde{T}}_{l_{1} \dots}^{k^{1} \dots} f_{k_{1}} \otimes \dots

Invariance of the Sum Despite Changing Terms

Although both the individual basis tensor products and the individual components change when the basis changes, the complete sum they form remains equal to the same abstract tensor T, since the two changes are inverse to one another and cancel exactly when the full expansion is reconstituted.

Basis Description of Special Tensors

The Identity Tensor

The type (1, 1) identity tensor, corresponding to the identity linear map on V, has the basis description δ^i_j e_i ⊗ e^j, where δ^i_j is the Kronecker delta, equal to 1 when i = j and 0 otherwise; this basis description is identical in every basis, reflecting the fact that the identity map does not depend on the coordinates used to represent it.

A Basis Vector's Own Description

Each basis vector e_k has the trivially simple basis description consisting of the single term e_k itself, with components equal to 1 in position k and 0 elsewhere, illustrating that the basis vectors serve as the simplest possible basis descriptions within the space they generate.

Diagrammatic Summary

The diagram shows a rank-two tensor described relative to a basis as a sum of basis tensor products, each one weighted by the corresponding component, with the full basis description recovering the abstract tensor once every term is added together.