Theorems · Theorem · linear algebra

LinearMap.finrank_range_add_finrank_ker

∀ {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] (f : V →ₗ[K] V₂),
  Module.finrank K ↥f.range + Module.finrank K ↥f.ker = Module.finrank K V

rank-nullity theorem : the dimensions of the kernel and the range of a linear map add up to the dimension of the source space.

Defined in

Mathlib.LinearAlgebra.FiniteDimensional.Lemmas

Cited by

14 results in Mathlib

Foundations

Depth 117 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DivisionRingAddCommGroupModuleAddCommGroupModuleFiniteDimensional