# 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.