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 $$x \in G: x = \sum_{k=1}^n \lambda_k a_k$$.

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, 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.