Definition:Coordinate System

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Definition

Let $R$ be a ring with unity.

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

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

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


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


Coordinate

Let $\sequence {a_n}$ 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 $\sequence {a_n}$.


Origin

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.


Also see

  • Results about coordinate systems can be found here.


Sources