Theorems · Theorem · linear algebra
LinearMap.range_lt_top_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.range < ⊤If the determinant of a map vanishes, then the map is not onto.
- Defined in
- Mathlib.LinearAlgebra.Determinant
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 127 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites20
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
- Top.topstatement and proof · cited by 9,680
- Submodulestatement · cited by 7,192
- MonoidHomstatement · cited by 3,629
- IsDomainstatement and proof · cited by 2,196
- LinearMap.compproof · cited by 1,642
- MulZeroClass.zero_mulproof · cited by 1,625
Cited by1
Results whose statement or proof uses this declaration.
- MeasureTheory.Measure.addHaar_image_linearMapproof · cited by 4