Theorems · Theorem · linear algebra

LinearMap.injective_iff_surjective_of_finrank_eq_finrank

∀ {K : Type u} {V : Type v} [inst : DivisionRing K] [inst_1 : AddCommGroup V] [inst_2 : Module K V] {V₂ : Type v'}
  [inst_3 : AddCommGroup V₂] [inst_4 : Module K V₂] [FiniteDimensional K V] [FiniteDimensional K V₂],
  Module.finrank K V = Module.finrank K V₂ → ∀ {f : V →ₗ[K] V₂}, Function.Injective ⇑f ↔ Function.Surjective ⇑f

Defined in

Mathlib.LinearAlgebra.FiniteDimensional.Lemmas

Cited by

1 results in Mathlib

Foundations

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

Assumes

DivisionRingAddCommGroupModuleAddCommGroupModuleFiniteDimensionalFiniteDimensional