Definition:Coordinate System/Coordinate

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Let $\sequence {a_n}$ be a coordinate system of a unitary $R$-module $G$.

Let $\ds x \in G: x = \sum_{k \mathop = 1}^n \lambda_k a_k$.

The scalars $\lambda_1, \lambda_2, \ldots, \lambda_n$ can be referred to as the coordinates of $x$ relative to $\sequence {a_n}$.

Elements of Ordered Pair

Let $\tuple {a, b}$ be an ordered pair.

The following terminology is used:

$a$ is called the first coordinate
$b$ is called the second coordinate.

This definition is compatible with the equivalent definition in the context of Cartesian coordinate systems.

Also known as

Coordinates of $x$ relative to $\sequence {a_n}$ are also known as coordinates of $x$ with respect to $\sequence {a_n}$.

Also denoted as

It is usual to use the subscript technique to denote the coordinates where $n$ is large or unspecified:

$\tuple {x_1, x_2, \ldots, x_n}$

However, note that some texts (often in the fields of physics and mechanics) prefer to use superscripts:

$\tuple {x^1, x^2, \ldots, x^n}$

While this notation is documented here, its use is not endorsed by $\mathsf{Pr} \infty \mathsf{fWiki}$ because:

there exists the all too likely subsequent confusion with notation for powers
one of the philosophical tenets of $\mathsf{Pr} \infty \mathsf{fWiki}$ is to present a system of notatiion that is as completely consistent as possible.

Historical Note

The words coordinate and coordinates entered the mathematical mainstream via the works of Gottfried Wilhelm von Leibniz, who may well have coined them.

Linguistic Note

It's an awkward word coordinate.

It really needs a hyphen in it to emphasise its pronunciation (loosely and commonly: coe-wordinate), and indeed, some authors spell it co-ordinate.

However, this makes it look unwieldy.

An older spelling puts a diaeresis indication symbol on the second "o": coördinate.

But this is considered archaic nowadays and few sources still use it.