2.4.3 Tensor Vector Element Coordinate Form

The Tensor Vector Element Coordinate Form expresses tensor components in a coordinate system, linking abstract algebra to spatial representation through indexed notation.

Tensor Vector Element Coordinate Form is the explicit expression of a tensor as a linear combination of basis tensors, with scalar coefficients given by the tensor's components, written using indexed notation and, conventionally, the Einstein summation convention. It is the working notation in which most computations with tensors are actually carried out, translating the abstract multilinear-map description of an element into an explicit finite formula built from basis vectors, dual basis covectors, and a labeled array of numbers.

Deriving the Coordinate Form

Setting

Let $V$ be a finite-dimensional vector space of dimension $n$ over a field $F$ , with basis $e_{1}, \dots, e_{n}$ and dual basis $e^{1}, \dots, e^{n}$ satisfying $e^{i} (e_{j}) = δ_{j}^{i}$ . A tensor $T \in T_{s}^{r} (V)$ has components defined by evaluating it on basis inputs:

T_{j_{1} \dots j_{s}}^{i_{1} \dots i_{r}} = T (e^{i_{1}}, \dots, e^{i_{r}}, e_{j_{1}}, \dots, e_{j_{s}})

The Coordinate Expression

Multilinearity guarantees that $T$ is fully recovered from these components through the expansion

T = T_{j_{1} \dots j_{s}}^{i_{1} \dots i_{r}} e_{i_{1}} \otimes \dots \otimes e_{i_{r}} \otimes e^{j_{1}} \otimes \dots \otimes e^{j_{s}}

with summation implied over every repeated index by the Einstein summation convention, ranging from $1$ to $n$ .

Index Conventions

Upper and Lower Placement

Indices corresponding to vector arguments of $T$ are written as lower indices on the components and paired with upper-indexed dual basis covectors $e^{j}$ ; indices corresponding to covector arguments are written as upper indices on the components and paired with lower-indexed basis vectors $e_{i}$ . This crisscross placement encodes, at a glance, which factor of the basis tensor each index refers to.

Free Versus Summed Indices

An index that appears exactly once in a term is a free index, ranging implicitly over all admissible values and indicating that the equation holds for each such value separately. An index that appears exactly twice, once upper and once lower, is a summed, or dummy, index, and the Einstein convention instructs that a sum over its full range is implicit even though no summation sign is written.

Component Access via Evaluation

Recovering a Single Component

The coordinate form is consistent with the defining evaluation formula: substituting $e_{k_{1}}$ and its counterparts into the expanded sum, only the term whose basis tensor matches survives, by orthogonality of the dual pairing, recovering exactly

T (e^{k_{1}}, \dots, e_{l_{1}}, \dots) = T_{l_{1} \dots}^{k_{1} \dots}

confirming that the coordinate form is not merely an approximation of $T$ but an exact reconstruction.

Coordinate Form of Sums and Scalar Multiples

Addition in Coordinates

Given the coordinate forms of two tensors of the same type, their sum has the coordinate form obtained by adding corresponding components, since the underlying basis tensors match term by term:

S + T = (S_{j_{1} \dots}^{i_{1} \dots} + T_{j_{1} \dots}^{i_{1} \dots}) e_{i_{1}} \otimes \dots \otimes e^{j_{1}} \otimes \dots

Scalar Multiples in Coordinates

Similarly, the coordinate form of $α T$ is obtained by multiplying every component by $α$ , leaving the basis tensors themselves unchanged, since scalar action distributes across the sum defining the coordinate expansion.

Coordinate Form Under Change of Basis

Transformation Law

If ${\tilde{e}}^{i}$ is a second basis related to the first by a matrix $A_{i}^{k}$ , the coordinate form of $T$ relative to the new basis has components related to the original components by the standard tensor transformation law, built from $A_{i}^{k}$ for each lower index and its inverse for each upper index. The two coordinate forms, though built from different basis tensors and different component arrays, expand to the same underlying element $T$ .

Invariance of the Full Expression

While individual components change under a change of basis, the full coordinate expression, components multiplied by their corresponding basis tensors and summed, is invariant, since it always reconstructs the same basis-independent multilinear map.

Practical Role of the Coordinate Form

Computation

The coordinate form is the format in which tensor operations, addition, scalar action, tensor products, and contractions, are carried out in explicit calculations, since it reduces every operation to indexed arithmetic on finite arrays of scalars governed by the Einstein summation convention.

Bridging Abstract and Concrete Descriptions

By expressing an intrinsically defined multilinear map as a finite, explicit sum over a chosen basis, the coordinate form provides the concrete handle through which the abstract element structure of a tensor space is manipulated in practice, without requiring direct reference to the space of all possible input tuples.