Theorems · Theorem · geometry

weightedVSub_mem_vectorSpan_pair

∀ {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 →
    ∀ {w w₁ w₂ : ι → k} {s : Finset ι},
      ∑ i ∈ s, w i = 0 →
        ∑ i ∈ s, w₁ i = 1 →
          ∑ i ∈ s, w₂ i = 1 →
            ((s.weightedVSub p) w ∈
                vectorSpan k {(Finset.affineCombination k s p) w₁, (Finset.affineCombination k s p) w₂} ↔
              ∃ r, ∀ i ∈ s, w i = r * (w₁ i - w₂ i))

Given an affinely independent family of points, a weighted subtraction lies in the vectorSpan of two points given as affine combinations if and only if it is a weighted subtraction with weights a multiple of the difference between the weights of the two points.

Defined in

Mathlib.LinearAlgebra.AffineSpace.Independent

Cited by

1 results in Mathlib

Foundations

Depth 58 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupModuleAddTorsor