Theorems · Definition · linear algebra

Subspace.quotEquivAnnihilator

{K : Type u_1} →
  {V : Type u_2} →
    [inst : Field K] →
      [inst_1 : AddCommGroup V] →
        [inst_2 : Module K V] →
          [FiniteDimensional K V] → (W : Subspace K V) → (V ⧸ W) ≃ₗ[K] ↥(Submodule.dualAnnihilator W)

The quotient by a subspace is isomorphic to its dual annihilator.

Defined in: Mathlib.LinearAlgebra.Dual.Lemmas
Cited by: 1 results in Mathlib
Foundations: Depth 120 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Field AddCommGroup Module FiniteDimensional

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Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Modulestatement and proof · cited by 20,661
RingHom.idstatement and proof · cited by 18,349
AddCommGroupstatement and proof · cited by 12,871
Fieldstatement and proof · cited by 7,404
Submodulestatement · cited by 7,192
LinearEquivstatement and proof · cited by 3,317
HasQuotient.Quotientstatement and proof · cited by 2,301
FiniteDimensionalstatement and proof · cited by 1,854
LinearMap.rangeproof · cited by 893
Module.Dualstatement and proof · cited by 583
LinearEquiv.transproof · cited by 298
Submodule.dualAnnihilatorstatement · cited by 77

Cited by1

Results whose statement or proof uses this declaration.

Subspace.finrank_add_finrank_dualAnnihilator_eqproof · cited by 3