Theorems · Theorem · linear algebra

LinearEquiv.mem_dilatransvections_iff_rank

∀ {V : Type u_2} [inst : AddCommGroup V] {K : Type u_3} [inst_1 : DivisionRing K] [inst_2 : Module K V] {e : V ≃ₗ[K] V},
  e ∈ LinearEquiv.dilatransvections K V ↔ Module.rank K ↥(↑e - LinearMap.id).range ≤ 1

Over a division ring, dilatransvections correspond to linear equivalences e such that the linear map e - id has rank at most 1. See also LinearEquiv.mem_dilatransvections_iff_finrank.

Defined in

Mathlib.LinearAlgebra.Transvection.Basic

Cited by

2 results in Mathlib

Foundations

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

Assumes

AddCommGroupDivisionRingModule