Theorems · Theorem · linear algebra
LinearMap.trace_id
∀ (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], (LinearMap.trace R M) LinearMap.id = ↑(Module.finrank R M)
The trace of the identity endomorphism is the dimension of the free module.
- Defined in
- Mathlib.LinearAlgebra.Trace
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 121 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
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 · cited by 18,349
- CommRingstatement and proof · cited by 17,173
- AddCommGroupstatement and proof · cited by 12,871
- LinearMapstatement · cited by 10,215
- Module.finrankstatement and proof · cited by 1,770
- Module.Finitestatement and proof · cited by 1,032
- Module.Endproof · cited by 774
- LinearMap.idstatement · cited by 625
- Module.Freestatement and proof · cited by 597
- LinearMap.tracestatement and proof · cited by 87
Cited by4
Results whose statement or proof uses this declaration.
- LinearMap.IsProj.traceproof · cited by 3
- LieModule.trace_toEnd_genWeightSpaceproof · cited by 3
- LieModule.traceForm_genWeightSpace_eqproof · cited by 1
- LieAlgebra.hasTrivialRadical_of_isIrreducible_of_isFaithfulproof · cited by 0