Definition:Coordinate System/Coordinates on Affine Space

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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 $\mathcal R$.