Definition:Coordinate System/Coordinate
Definition
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 notation that is as completely consistent as possible.
Also see
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.
Sources
- 1921: C.E. Weatherburn: Elementary Vector Analysis ... (previous) ... (next): Chapter $\text I$. Addition and Subtraction of Vectors. Centroids: Components of a Vector: $7$. The unit vectors $\mathbf i$, $\mathbf j$, $\mathbf k$
- 1964: Steven A. Gaal: Point Set Topology: Introduction to Set Theory: $3$. The Axiom of Choice and Its Equivalents
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $7$: Vector Spaces: $\S 35$. Coordinates
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): coordinate
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): coordinate
- 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $6$: Curves and Coordinates: Fermat
- For a video presentation of the contents of this page, visit the Khan Academy.