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 $\Theta_{\mathcal R} : k^n \to \mathcal E$ defined by
 * $\displaystyle \Theta_{\mathcal R}\left(\lambda_1,\ldots,\lambda_n\right) = p_0 + \sum_{i = 1}^n \lambda_ie_i$

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

Proof
First we show that $\Theta_{\mathcal R}$ is surjective.

Indeed, if $p \in \mathcal E$, 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 we have

Now let us show that $\Theta_{\mathcal R}$ is injective.

Suppose we have
 * $\Theta_{\mathcal R}\left(\lambda_1,\ldots,\lambda_n\right) = \Theta_{\mathcal R}\left(\mu_1,\ldots,\mu_n\right)$

That is,
 * $\displaystyle p_0 + \sum_{i = 1}^n \lambda_ie_i = p_0 + \sum_{i = 1}^n \mu_ie_i$

Then by $(3)$ of Properties of Affine Spaces this means that
 * $\displaystyle \sum_{i = 1}^n \lambda_ie_i = \sum_{i = 1}^n \mu_ie_i$

Now by Expression of Vector as Linear Combination from Basis is Unique this shows that $\lambda_i = \mu_i$ for $i = 1,\ldots,n$.

This concludes the proof.