Definition:Coordinate System/Coordinate

Coordinate System
Let $\left \langle {a_n} \right \rangle$ be an ordered basis of a unitary $R$-module $G$.

Then $\left \langle {a_n} \right \rangle$ 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 $\displaystyle x \in G: x = \sum_{k=1}^n \lambda_k a_k$.

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

Origin
The origin of a coordinate system is the zero vector.

Comment
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.