Affine Coordinates are Well-Defined

Theorem
Let $\mathcal E$ be an affine space with difference space $V$ over a field $k$.

Let $\mathcal R = \left({p_0, e_1, \ldots, e_n}\right)$ be an affine frame in $\mathcal E$.

Define a mapping $\Theta_{\mathcal R} : k^n \to \mathcal E$ by:
 * $\displaystyle \Theta_\mathcal R \left({\lambda_1, \ldots, \lambda_n}\right) = p_0 + \sum_{i \mathop = 1}^n \lambda_i e_i$

Then $\Theta_\mathcal R$ is a bijection.

Proof of Surjection
Let $p \in \mathcal E$.

Let $v = p - p_0 \in V$.

Let $\left({\lambda_1, \ldots, \lambda_n}\right)$ be coordinates of $v$ in the basis $\left({e_1, \ldots, e_n}\right)$.

Then:

thus demonstrating that $\Theta_\mathcal R$ is a surjection.

Proof of Injection
Let:
 * $\Theta_\mathcal R \left({\lambda_1, \ldots, \lambda_n}\right) = \Theta_\mathcal R \left({\mu_1, \ldots, \mu_n}\right)$

That is:
 * $\displaystyle p_0 + \sum_{i \mathop = 1}^n \lambda_i e_i = p_0 + \sum_{i \mathop = 1}^n \mu_i e_i$

Then by $(3)$ of Properties of Affine Spaces:
 * $\displaystyle \sum_{i \mathop = 1}^n \lambda_i e_i = \sum_{i \mathop = 1}^n \mu_i e_i$

By Expression of Vector as Linear Combination from Basis is Unique:
 * $\lambda_i = \mu_i$

for $i = 1, \ldots, n$.

Hence the result.