2.10.1 Tensor Coordinate Vector Component List

Tensor Coordinate Vector Component List is the ordered listing of individual scalar entries that together make up a coordinate vector, where each entry, or component, corresponds to the coefficient attached to one specific basis vector in the linear combination that reconstructs the represented vector. The component list is the concrete data structure through which a coordinate vector is stored, indexed, and manipulated in tensor computations.

Formal Statement

Components as Indexed Entries

A coordinate vector of length n is fully described by its list of components, each identified by a positional index running from the first basis vector to the last.

Individual Component Extraction

Each component can be individually extracted from the list by referring to its index, giving direct access to the coefficient corresponding to the basis vector at that position.

Structural Properties of the Component List

Fixed Length Determined by Dimension

The number of components in the list equals the dimension of the vector space, since there is exactly one component for each basis vector, linking the component list directly to dimension coordinate length.

Every Component Belongs to the Field

Each entry in the component list is drawn from the same field over which the vector space is defined, so the component list can be understood as an element of the field raised to a power equal to the dimension.

Operations on Component Lists

Entrywise Combination Under Vector Operations

Vector addition and scalar multiplication translate into entrywise operations on the component list, so that combining two coordinate vectors or scaling one is performed component by component using ordinary field arithmetic.

Zero Component List for the Zero Vector

The component list consisting entirely of zero entries corresponds to the zero vector of the space, since the trivial linear combination with every coefficient equal to zero always produces the zero vector.

Role in Tensor Construction

Building Blocks for Tensor Index Structures

Component lists from several vector spaces serve as the building blocks for the indexed component structures used to describe tensors, where each tensor component is obtained by combining one entry from each contributing coordinate vector's component list.

Practical Storage Format

Because component lists are simply ordered collections of field elements, they translate directly into the arrays or lists used in software and mathematical notation to store and process vectors and tensors numerically.

Summary of Key Properties

Concrete Enumeration of Coordinate Data

Tensor Coordinate Vector Component List provides the explicit, position-by-position enumeration of the scalar data that a coordinate vector carries, making the abstract coordinate vector directly usable.

Direct Link to Dimension and Basis Order

The length and meaning of the component list are entirely determined by the dimension of the space and the order of the fixed basis, tying this structure closely to both dimension structure and coordinate vector ordering.