Theorems · Theorem · linear algebra
Module.Basis.sum_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]
[inst_3 : Fintype ι] (b : Module.Basis ι R M) (u : M), ∑ i, (b.repr u) i • b i = u- Defined in
- Mathlib.LinearAlgebra.Basis.Defs
- Cited by
- 29 results in Mathlib
- Foundations
- Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound
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 · 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
- Fintypestatement and proof · cited by 7,736
- Finsuppstatement · cited by 5,255
- Finset.sumstatement · cited by 5,195
- Finset.univstatement · cited by 3,473
- LinearEquivstatement · cited by 3,317
- Module.Basisstatement and proof · cited by 1,477
- Module.Basis.reprstatement · cited by 498
Cited by29
Results whose statement or proof uses this declaration.
- OrthonormalBasis.sum_reprproof · cited by 7
- LinearMap.BilinForm.dualSubmodule_span_of_basisproof · cited by 7
- IsLocalFrameOn.coeff_sum_eqproof · cited by 5
- Submodule.basis_of_pid_auxproof · cited by 2
- ZSpan.norm_fract_leproof · cited by 2
- ZSpan.fundamentalDomain_ae_parallelepipedproof · cited by 2
- Module.Basis.sum_toMatrix_smul_selfproof · cited by 2
- NumberField.Units.exist_unique_eq_mul_prodproof · cited by 2
- apply_eq_dotProduct_toMatrix₂_mulVecproof · cited by 2
- AddChar.exists_apply_ne_zeroproof · cited by 1
- IsDedekindDomain.range_sup_range_eq_top_of_isCoprime_differentIdealproof · cited by 1
- IsLocalRing.basisQuotient_reprproof · cited by 1