Motivation: I would like to prove characterizations of the affine basis property without using the notion of affine combinations.
I am working with the following definitions.
Definition: For any affine space $\mathcal{A} = (A, V, +)$ and any $\emptyset \neq X \subseteq A$, the intersection of all $X \subseteq B \subseteq A$ such that $B$ induces an affine subspace of $\mathcal{A}$ is called the affine span of $X$ and is denoted by $\text{aff} X$.
Definition: Let $\mathcal{A} = (A, V, +)$ an affine space and $X \subseteq A$. $X$ is called an affine spanning set of $\mathcal{A}$ and is said to affinely span $\mathcal{A}$ if and only if $X \neq \emptyset$ and $\mathrm{aff} X = A$. $X$ is said to be affinely independent if and only if $X \neq \emptyset$ and for any $\emptyset \neq Y \subset X$ we have $\text{aff} Y \subset \text{aff} X$. $X$ is called an affine basis of $\mathcal{A}$ if and only if $X$ is affinely independent and an affine spanning set of $\mathcal{A}$.
How can I prove the following proposition from these definitions and more elementary properties of affine spaces? (Rough and partial sketches will be appreciated, as well as references to textbooks).
Proposition: For any affine space $\mathcal{A} = (A, V, +)$ and any $X \subseteq A$ the following statements are equivalent:
- $X$ is a maximal affinely independent subset of $A$.
- $X$ is a minimal affine spanning set of $\mathcal{A}$.
- $X$ is an affine basis of $\mathcal{A}$.
For your affine span, just take $\operatorname{span}_{\text{affine}} X = X + \operatorname{span}_{\text{vect}}(X-X)$ or $\{x + \lambda (y-x),\, x ,y \in X, \, \lambda \in k\}$
For affine independant, just say that $(x_0,\dots, x_n)$ is affinity independent if $(x_1-x_0,\dots,x_n-x_0)$ is free.
Then it is equivalent to have maximal affine independent, and minimal generating (you can easily see that it is generating if $\operatorname{span}_{\text{vect}} (x_1-x_0,\dots,x_n-x_0)$ is. This gives you the property "is an affine basis", and is equivalent to $(x_1-x_0,\dots,x_n-x_0)$ is a basis of the vector space.
The technique here is called "linearization".