Theorems · Theorem · geometry

affineIndependent_set_iff_linearIndependent_vsub

∀ (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] {s : Set P} {p₁ : P},
  p₁ ∈ s → ((AffineIndependent k fun p => ↑p) ↔ LinearIndependent k fun (v : ↑((fun p => p -ᵥ p₁) '' (s \ {p₁}))) => ↑v)

A set is affinely independent if and only if the differences from a base point in that set are linearly independent.

Defined in

Mathlib.LinearAlgebra.AffineSpace.Independent

Cited by

3 results in Mathlib

Foundations

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

Assumes

RingAddCommGroupModuleAddTorsor