Theorems · Theorem · geometry

eq_affineCombination_of_mem_affineSpan

∀ {ι : Type u_1} {k : Type u_2} {V : Type u_3} {P : Type u_4} [inst : Ring k] [inst_1 : AddCommGroup V]
  [inst_2 : Module k V] [inst_3 : AddTorsor V P] {p1 : P} {p : ι → P},
  p1 ∈ affineSpan k (Set.range p) → ∃ s w, ∑ i ∈ s, w i = 1 ∧ p1 = (Finset.affineCombination k s p) w

A point in the affineSpan of an indexed family is an affineCombination with sum of weights 1. See also eq_affineCombination_of_mem_affineSpan_of_fintype.

Defined in

Mathlib.LinearAlgebra.AffineSpace.Combination

Cited by

4 results in Mathlib

Foundations

Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupModuleAddTorsor