9.12.3 Tensor Basis Expansion Summation Form

The Tensor Basis Expansion Summation Form expresses tensor components through a basis, enabling systematic manipulation and computation in multilinear algebra.

Tensor Basis Expansion Summation Form is the explicit representation of a tensor as a sum of basis tensor products, each weighted by its corresponding component. Instead of treating a tensor as an abstract multilinear map, this form writes the tensor concretely as a linear combination of tensor-product basis elements, with the components acting as the coefficients of that combination.

Structure of the Expansion

General Form

For a tensor of type (p, q), the expansion is written as a sum over all combinations of basis vectors and dual basis covectors, indexed by the contravariant indices i₁, ..., i_p and the covariant indices j₁, ..., j_q.

T = \sum_{i_{1}, \dots, j_{q}} 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}}

Here the symbols e with a lower index denote basis vectors of the underlying vector space, and the symbols e with an upper index denote the dual basis covectors. The components carry upper indices for contravariant slots and lower indices for covariant slots, matching the position of the basis elements they multiply.

Einstein Summation Convention

Under the Einstein summation convention, the explicit summation symbol is dropped whenever an index appears once as a superscript and once as a subscript within the same term, since repetition of an index in this paired position is understood to imply summation over its full range.

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

This compact notation is the standard working form used throughout tensor algebra, since writing out summation symbols for every contraction quickly becomes impractical in expressions with several indices.

Role of the Components

Coefficients of the Expansion

The component array associated with a tensor is not an independent object; it is defined precisely as the set of coefficients required to reconstruct the tensor from a chosen basis through the summation form. Changing the basis changes the numerical values of the components, but the sum as a whole, the tensor itself, remains invariant.

Extraction of Components

A component of a tensor can be recovered by pairing the tensor with the appropriate dual basis elements and basis vectors, since the basis and dual basis are constructed to satisfy the duality relation in which a dual basis covector applied to a basis vector yields one when the indices match and zero otherwise. Applying this pairing to the summation form isolates the single term corresponding to the chosen indices.

Basis Dependence and Invariance

Change of Basis

When the underlying basis is changed, the basis vectors transform under one rule and the dual basis covectors transform under the inverse rule. The summation form guarantees that these two opposite transformations cancel exactly against the transformation of the components, so the total object described by the sum does not depend on which basis was used to expand it.

Consistency Requirement

Because of this cancellation, the summation form is the mechanism that ties together three otherwise separate pieces of data: the choice of basis vectors, the choice of dual basis covectors, and the array of components. None of the three pieces alone constitutes the tensor; only the complete summation expresses the tensor unambiguously.

Practical Use

Contraction and Manipulation

Writing a tensor in summation form makes operations such as contraction, raising and lowering of indices, and multiplication with other tensors straightforward, since these operations reduce to algebraic manipulation of the indices and coefficients while the underlying basis tensor products follow fixed multiplication rules.

Finite Sums in Finite Dimensions

In a finite-dimensional vector space, the summation form always consists of a finite number of terms, equal to the product of the dimension raised to the power of the total number of indices. This finiteness is what allows the tensor to be stored, computed with, and manipulated as a concrete array of numbers once a basis has been fixed.