Theorems · Theorem · geometry

affineIndependent_iff_indicator_eq_of_affineCombination_eq

∀ (k : Type u_1) {V : Type u_2} {P : Type u_3} [inst : Ring k] [inst_1 : AddCommGroup V] [inst_2 : Module k V]
  [inst_3 : AddTorsor V P] {ι : Type u_4} (p : ι → P),
  AffineIndependent k p ↔
    ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),
      ∑ i ∈ s1, w1 i = 1 →
        ∑ i ∈ s2, w2 i = 1 →
          (Finset.affineCombination k s1 p) w1 = (Finset.affineCombination k s2 p) w2 →
            (↑s1).indicator w1 = (↑s2).indicator w2

A family is affinely independent if and only if any affine combinations (with sum of weights 1) that evaluate to the same point have equal Set.indicator.

Defined in

Mathlib.LinearAlgebra.AffineSpace.Independent

Cited by

4 results in Mathlib

Foundations

Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupModuleAddTorsor