Definition:Coordinate System/Coordinates on Affine Space

Definition
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.

By Affine Coordinates Well Defined there exists a unique ordered tuple $\left(\lambda_1,\ldots,\lambda_n\right) \in k^n$ such that
 * $\displaystyle p = p_0 + \sum_{i = 1}^n \lambda_i e_i$

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