Theorems · Theorem · linear algebra
Matrix.mulVec_injective_of_det_mem_nonZeroDivisors
∀ {m : Type u_1} {R : Type u_2} [inst : CommRing R] [inst_1 : Fintype m] [inst_2 : DecidableEq m] {M : Matrix m m R},
M.det ∈ nonZeroDivisors R → Function.Injective M.mulVecSee also Matrix.mulVec_injective_iff_isUnit when working over a field.
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 103 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRingFintypeDecidableEq
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Fintypestatement and proof · cited by 7,736
- Matrixstatement and proof · cited by 4,303
- Submonoidstatement · cited by 3,086
- sub_selfproof · cited by 996
- nonZeroDivisorsstatement and proof · cited by 895
- Matrix.detstatement and proof · cited by 665
- sub_eq_zeroproof · cited by 407
- Matrix.mulVecstatement and proof · cited by 267
- Matrix.mulVec_subproof · cited by 3
- Matrix.eq_zero_of_det_mem_nonZeroDivisors_of_mulVec_eq_zeroproof · cited by 1
Cited by3
Results whose statement or proof uses this declaration.
- Matrix.rank_mul_eq_right_of_det_mem_nonZeroDivisorsproof · cited by 2
- Matrix.rank_of_det_mem_nonZeroDivisorsproof · cited by 1
- Matrix.mulVec_injective_of_det_ne_zeroproof · cited by 0