Definition:Coordinate System

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Let $R$ be a ring with unity.

Let $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$ be an ordered basis of a free unitary $R$-module $G$.

Then $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$ can be referred to as a coordinate system.

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 $\EE$ be an affine space of dimension $n$ over a field $k$.

Let $\RR = \tuple {p_0, e_1, \ldots, e_n}$ be an affine frame in $\EE$.

Let $p \in \EE$ be a point.

Since Affine Coordinates are Well-Defined, there exists a unique ordered tuple $\tuple {\lambda_1, \ldots, \lambda_n} \in k^n$ such that:

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

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


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 $\sequence {a_n}$.


The origin of a coordinate system is the zero vector.

In the $x y$-plane, it is the point:

$O = \tuple {0, 0}$

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

$O = \underbrace {\tuple {0, 0, \ldots, 0} }_{\text{$n$ coordinates} }$

Thus it is the point where the axes cross over each other.


In physics and applied mathematics, it is usual for the coordinate system under discussion to be considered as superimposed on ordinary space of $3$ dimensions.

Also see

  • Results about coordinate systems can be found here.