Definition:Coordinate System
Definition
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 $R$-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 $\mathcal E$ 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
- 1964: D.E. Rutherford: Classical Mechanics (3rd ed.) ... (previous) ... (next): Chapter $\text I$: Kinematics: $1$. Space and Time
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): $\S 27$
- 1974: Robert Gilmore: Lie Groups, Lie Algebras and Some of their Applications ... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $4$. LINEAR VECTOR SPACE