Matroid Induced by Affine Independence is Matroid

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Theorem

Let $\R^n$ be the $n$-dimensional real Euclidean space.

Let $S = \set{x_1, \dots, x_r}$ be a finite subset of $\R^n$.

Let $\struct{S, \mathscr I}$ be the matroid induced by affine independence on $S$.

That is, $\mathscr I$ is the set of affinely independent subsets of $S$.


Then $\struct{S, \mathscr I}$ is a matroid.

Proof

It needs to be shown that $\mathscr I$ satisfies the matroid axioms $(I1)$, $(I2)$ and $(I3)$.

Matroid Axiom $(I1)$


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Matroid Axiom $(I2)$


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Matroid Axiom $(I3)$


This theorem requires a proof.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
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Sources

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