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OpenML Research: Matroid Theory Section 1.1

Reference Literature

Formal Definition

A matroid M is an ordered pair (E, Ι) consisting of a finite set Eand a collection Ι of subsets of E having these three properties:

(I1) \emptyset \in Ι (I2) If I \in Ι and J \subseteq I, then J \in Ι. (I3) If I₁, I₂ \in Ι and |I₁| < |I₂|, then there is e \in I₂ - I₁ s.t. I₁ \cup {e} \in Ι.

Active Experiments

Set I₁ Independent

Size: 0 | {∅}

Set I₂ Independent

Size: 0 | {∅}

Conceptual Mapping

When Oxley writes I ∈ Ι, he is referring to the "Independence Club."The ground set E is your universe of choices—here, vectors v₁ through v₅. If you pick v₁, v₂, and v₃ and they don't collapse into a plane, that set{v₁, v₂, v₃} belongs to the family of independent sets.

The subset notation J ⊆ I represents a smaller selection from your chosen plan.Axiom (I2) is the hereditary property of freedom: if a set of actions is unconstrained, any subset remains so. Geometrically, if a 3D volume is unconstrained, the 2D sheets and 1D lines making it up are also unconstrained. You only lose independence by adding redundancy.

Linear dependence is the state of a set that fails to join Ι. It implies a hidden constraint or redundancy. If you select v₁, v₂, and v₄, the set is dependent because v₄ lies in the same 2D sheet as v₁ and v₂. It adds no new agency or directional freedom; it is redundant choice.