Theorems · Theorem · geometry

linearIndependent_set_iff_affineIndependent_vadd_union_singleton

∀ (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 V},
  (∀ v ∈ s, v ≠ 0) → ∀ (p₁ : P), (LinearIndependent k fun (v : ↑s) => ↑v) ↔ AffineIndependent k fun p => ↑p

A set of nonzero vectors is linearly independent if and only if, given a point p₁, the vectors added to p₁ and p₁ itself are affinely independent.

Defined in

Mathlib.LinearAlgebra.AffineSpace.Independent

Cited by

2 results in Mathlib

Foundations

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

Assumes

RingAddCommGroupModuleAddTorsor