Theorems · Theorem · geometry

AffineIndependent.vectorSpan_eq_of_le_of_card_eq_finrank_add_one

∀ {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},
  AffineIndependent k p →
    ∀ {sm : Submodule k V} [FiniteDimensional k ↥sm],
      vectorSpan k (Set.range p) ≤ sm → Fintype.card ι = Module.finrank k ↥sm + 1 → vectorSpan k (Set.range p) = sm

If the vectorSpan of a finite affinely independent family lies in a submodule with dimension one less than its cardinality, it equals that submodule.

Defined in

Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Cited by

0 results in Mathlib

Foundations

Depth 120 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DivisionRingAddCommGroupModuleAddTorsorFintypeFiniteDimensional