1.2.10 Coordinate Definition

Coordinate Definition is the characterization of a coordinate of a vector, relative to a chosen basis, as the unique scalar coefficient that multiplies a corresponding basis vector in the linear combination expressing that vector, and, collectively, of the coordinates of a vector as the ordered set of all such coefficients. It specifies precisely what a coordinate is at the level of vector spaces, providing the elementary numerical building block from which tensor components, and the tensor transformation law governing them, are constructed.

Coordinates as Coefficients of a Linear Combination

Given a basis for a vector space, every vector in that space can be written, uniquely, as a linear combination of the basis vectors. The coordinates of the vector, relative to that basis, are the coefficients appearing in this linear combination, taken in the order corresponding to the chosen ordering of the basis vectors. Because the basis representation of any given vector is unique, once the basis is fixed, the coordinates of that vector are likewise uniquely determined, and can be treated as a well-defined numerical description of the vector.

The expression above expresses a vector as a linear combination of basis vectors, with the coefficients labeled by superscripts constituting the coordinates of the vector relative to this basis.

Coordinates Are Basis-Dependent

A vector, considered as an abstract object, exists independently of any basis, but its coordinates do not: they are defined only relative to a specific choice of basis, and the same vector will, in general, have entirely different coordinates when expressed relative to a different basis. This dependence is not a defect but a deliberate feature of the concept, since it is precisely this dependence that motivates the study of how coordinates change when the basis changes, culminating in the tensor transformation law that governs tensors of every rank.

Indexing Conventions

Within tensor algebra, coordinates of vectors are conventionally written with an upper index, called a contravariant index, reflecting how these coordinates change under a change of basis: when the basis vectors are transformed by a certain linear substitution, the coordinates transform by the inverse substitution, in a sense that is made precise by the tensor transformation law. This convention, distinguishing upper indices for vector-like coordinates from lower indices used for covector-like coordinates, is maintained consistently throughout tensor algebra to keep track of how each set of components behaves under a change of basis.

Coordinates of Higher-Rank Objects

The notion of a coordinate extends naturally beyond vectors. A covector, once a dual basis has been chosen, likewise has coordinates, conventionally written with a lower index. A tensor of higher rank, once a basis has been chosen for each vector and dual space involved, has coordinates indexed by multiple indices simultaneously, some upper and some lower depending on the tensor's type, with each index transforming according to the appropriate contravariant or covariant rule under a change of basis.

Why the Coordinate Definition Matters

Coordinates provide the essential bridge between the abstract, basis-independent objects of tensor algebra and the concrete numerical calculations required in applications. Every explicit computation involving vectors, covectors, or tensors — whether in linear algebra, differential geometry, or physics — proceeds by first fixing a basis, expressing the relevant objects in terms of their coordinates, carrying out the calculation using ordinary arithmetic on these numbers, and, where necessary, translating the result back into basis-independent terms. The precise definition of a coordinate, together with the transformation rule describing how coordinates change between bases, is what guarantees that this process yields results consistent with the underlying, basis-independent mathematical reality.