Definition:Coordinate

Definition

Elements of Ordered Pair

Let $\left({a, b}\right)$ 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 coordinates.

Some authors use the terms first component and second component instead.

Coordinate System

Let $\left \langle {a_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ be an ordered basis of a unitary $R$-module $G$.

Then $\left \langle {a_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ can be referred to as a coordinate system.

Coordinate

Let $\left \langle {a_n} \right \rangle$ be a coordinate system of a unitary $R$-module $G$.

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

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

Coordinate Function

Let $\left \langle {a_n} \right \rangle$ be a coordinate system of a unitary $R$-module $G$.

For each $x \in G$ let $x_1, x_2, \ldots, x_n$ be the coordinates of $x$ relative to $\left \langle {a_n} \right \rangle$.

Then for $i = 1, \ldots, n$ the mapping $f_i : G \to R$ defined by $f_i \left({x}\right) = x_i$ is called the $i$-th coordinate function on $G$ relative to $\left \langle {a_n} \right \rangle$.

Coordinates on Affine Space

Let $\mathcal E$ be an affine space of dimension $n$ over a field $k$.

Let $\mathcal R = \left(p_0,e_1,\ldots,e_n\right)$ be an affine frame in $\mathcal E$.

Let $p \in \mathcal E$ be a point.

By Affine Coordinates Well Defined there exists a unique ordered tuple $\left(\lambda_1,\ldots,\lambda_n\right) \in k^n$ such that

$\displaystyle p = p_0 + \sum_{i = 1}^n \lambda_i e_i$

The numbers $\lambda_1,\ldots,\lambda_n$ are the coordinates of $p$ in the frame $\mathcal R$.

Origin

The origin of a coordinate system is the zero vector.

In the $xy$-plane, it is the point:

$O = \left({0, 0}\right)$

and in general, in the Euclidean space $\R^n$:

$O = \underbrace{\left({0, 0, \ldots, 0}\right)}_{n \ \text{coordinates}}$

Linguistic Note

It's an awkward word coordinate. It really needs a hyphen in it to emphasise its pronounciation (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

• Steven A. Gaal: Point Set Topology (1964)... (previous)... (next): Introduction to Set Theory: $3$. The Axiom of Choice and Its Equivalents
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 7.35$
• Paul Halmos and Steven Givant: Introduction to Boolean Algebras (2008)... (previous)... (next): Appendix $\text{A}$: Set Theory: Families

• For a video presentation of the contents of this page, visit the Khan Academy.