Theorems · Theorem · linear algebra
Module.Basis.repr_symm_apply
∀ {ι : Type u_1} {R : Type u_3} {M : Type u_6} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
(b : Module.Basis ι R M) (v : ι →₀ R), b.repr.symm v = (Finsupp.linearCombination R ⇑b) v- Defined in
- Mathlib.LinearAlgebra.Basis.Defs
- Cited by
- 37 results in Mathlib
- Foundations
- Depth 81 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SemiringAddCommMonoidModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites17
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
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- LinearMapstatement · cited by 10,215
- Finsuppstatement and proof · cited by 5,255
- LinearEquivstatement · cited by 3,317
- Module.Basisstatement and proof · cited by 1,477
- LinearEquiv.symmstatement and proof · cited by 1,461
- Finsupp.singleproof · cited by 943
- Module.Basis.reprstatement and proof · cited by 498
Cited by37
Results whose statement or proof uses this declaration.
- Module.Basis.equivFun_symm_applyproof · cited by 22
- Module.Basis.equiv_applyproof · cited by 13
- Module.Basis.tensorProduct_applyproof · cited by 9
- Module.Basis.baseChange_applyproof · cited by 7
- Module.Basis.coe_repr_symmproof · cited by 7
- Algebra.FormallyUnramified.finite_of_freeproof · cited by 7
- DirectSum.IsInternal.collectedBasis_coeproof · cited by 6
- Algebra.Generators.CotangentSpace.compEquiv_symm_inrproof · cited by 4
- Algebra.Generators.CotangentSpace.fst_compEquivproof · cited by 4
- IsBaseChange.basis_applyproof · cited by 3
- Module.finitePresentation_of_free_of_surjectiveproof · cited by 3
- LinearMap.IsOrthoᵢ.not_isOrtho_basis_self_of_separatingLeftproof · cited by 3