Theorems · Theorem · linear algebra
LinearMap.bot_lt_ker_of_det_eq_zero
∀ {R : Type u_1} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : Module R M] [IsDomain R]
[Module.Free R M] {f : M →ₗ[R] M}, LinearMap.det f = 0 → ⊥ < f.kerIf the determinant of a map vanishes, then the map is not injective.
- Defined in
- Mathlib.LinearAlgebra.Determinant
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 130 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites36
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
- Submodulestatement · cited by 7,192
- Bot.botstatement · cited by 4,720
- MonoidHomstatement · cited by 3,629
- Finset.univproof · cited by 3,473
- Finset.sum_congrproof · cited by 2,323
- mul_commproof · cited by 2,262
Cited by3
Results whose statement or proof uses this declaration.
- LinearMap.det_eq_zero_iff_ker_ne_botproof · cited by 3
- LinearMap.hasEigenvalue_zero_tfaeproof · cited by 2
- det_ne_zero_of_sum_row_lt_diagproof · cited by 1