Theorems · Theorem · linear algebra
Module.Basis.linearCombination_repr
∀ {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) (x : M), (Finsupp.linearCombination R ⇑b) (b.repr x) = x- Defined in
- Mathlib.LinearAlgebra.Basis.Defs
- Cited by
- 26 results in Mathlib
- Foundations
- Depth 83 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.
Cites13
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
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- LinearMapstatement and proof · cited by 10,215
- Finsuppstatement and proof · cited by 5,255
- LinearEquivstatement · cited by 3,317
- Module.Basisstatement and proof · cited by 1,477
- Module.Basis.reprstatement and proof · cited by 498
- Finsupp.linearCombinationstatement · cited by 269
- LinearEquiv.symm_apply_applyproof · cited by 78
Cited by26
Results whose statement or proof uses this declaration.
- Module.Basis.extproof · cited by 63
- Algebra.FormallyUnramified.finite_of_freeproof · cited by 7
- Module.Basis.ext'proof · cited by 6
- Module.Basis.mem_span_iff_repr_memproof · cited by 6
- Module.Basis.dualBasis_reprproof · cited by 5
- IsBaseChange.basis_repr_comp_applyproof · cited by 3
- PowerBasis.aeval_minpolyGenproof · cited by 3
- Module.finitePresentation_of_free_of_surjectiveproof · cited by 3
- Module.Basis.mem_span_repr_supportproof · cited by 3
- LinearEquiv.mem_dilatransvections_iff_rankproof · cited by 2
- LinearMap.sum_repr_mul_repr_mulproof · cited by 2
- Module.Basis.toDual_apply_leftproof · cited by 2