Theorems · Theorem · geometry

affineIndependent_iff_finrank_vectorSpan_eq

∀ (k : Type u_1) {V : Type u_2} {P : Type u_3} {ι : Type u_4} [inst : DivisionRing k] [inst_1 : AddCommGroup V]
  [inst_2 : Module k V] [inst_3 : AddTorsor V P] [inst_4 : Fintype ι] (p : ι → P) {n : ℕ},
  Fintype.card ι = n + 1 → (AffineIndependent k p ↔ Module.finrank k ↥(vectorSpan k (Set.range p)) = n)

n + 1 points are affinely independent if and only if their vectorSpan has dimension n.

Defined in

Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Cited by

1 results in Mathlib

Foundations

Depth 116 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DivisionRingAddCommGroupModuleAddTorsorFintype