Definition:Coordinate System/Coordinates on Affine Space

Definition
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 $\RR$.