Matroid Induced by Linear Independence in Abelian Group is Matroid
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Theorem
Let $\struct{G, +}$ be a torsion-free Abelian group.
Let $\struct{G, +, \times}$ be the $\Z$-module associated with $G$.
Let $S$ be a finite subset of $G$.
Let $\struct{S, \mathscr I}$ be the matroid induced by linear independence in $G$ on $S$.
That is, $\mathscr I$ is the set of linearly 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 $(I3)$
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Sources
- 1976: Dominic Welsh: Matroid Theory ... (previous) ... (next) Chapter $1.$ $\S 3.$ Examples of Matroids