Theorems · Theorem · geometry

AffineIndependent.finrank_vectorSpan

∀ {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 → ∀ {n : ℕ}, Fintype.card ι = n + 1 → Module.finrank k ↥(vectorSpan k (Set.range p)) = n

The vectorSpan of a finite affinely independent family has dimension one less than its cardinality.

Defined in

Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Cited by

9 results in Mathlib

Foundations

Depth 118 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DivisionRingAddCommGroupModuleAddTorsorFintype