Theorems · Theorem · linear algebra

LinearMap.trace_eq_contract_apply

∀ (R : Type u_1) [inst : CommRing R] (M : Type u_2) [inst_1 : AddCommGroup M] [inst_2 : Module R M] [Module.Free R M]
  [Module.Finite R M] (x : TensorProduct R (Module.Dual R M) M),
  (LinearMap.trace R M) ((dualTensorHom R M M) x) = (contractLeft R M) x

Defined in: Mathlib.LinearAlgebra.Trace
Cited by: 5 results in Mathlib
Foundations: Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing AddCommGroup Module Module.Free Module.Finite

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Cites15

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

DFunLike.coestatement and proof · cited by 62,936
Modulestatement and proof · cited by 20,661
RingHom.idstatement and proof · cited by 18,349
CommRingstatement and proof · cited by 17,173
AddCommGroupstatement and proof · cited by 12,871
LinearMapstatement and proof · cited by 10,215
TensorProductstatement and proof · cited by 2,545
Module.Finitestatement and proof · cited by 1,032
Module.Freestatement and proof · cited by 597
Module.Dualstatement and proof · cited by 583
LinearMap.tracestatement and proof · cited by 87
LinearMap.comp_applyproof · cited by 76

Cited by5

Results whose statement or proof uses this declaration.

LinearMap.trace_comp_commproof · cited by 1
LinearMap.trace_tensorProductproof · cited by 1
LinearMap.trace_transposeproof · cited by 1
Module.Free.bijective_algebraMap_of_finrank_eq_oneproof · cited by 1
LinearMap.trace_prodMapproof · cited by 1