Theorems · Theorem · linear algebra
Matrix.separatingLeft_iff_forall_vecMul_eq_zero
∀ {m : Type u_1} {n : Type u_2} {R : Type u_3} [inst : CommSemiring R] {M : Matrix m n R} [inst_1 : Fintype m]
[inst_2 : Finite n], M.SeparatingLeft ↔ ∀ (v : m → R), Matrix.vecMul v M = 0 → v = 0- Cited by
- 2 results in Mathlib
- Foundations
- Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringFintypeFinite
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites17
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommSemiringstatement and proof · cited by 10,911
- Fintypestatement and proof · cited by 7,736
- Matrixstatement and proof · cited by 4,303
- Finitestatement and proof · cited by 3,029
- one_smulproof · cited by 1,374
- MulOppositeproof · cited by 1,135
- Pi.singleproof · cited by 518
- Matrix.mulVecproof · cited by 267
- Fintype.ofFiniteproof · cited by 255
- dotProductproof · cited by 194
- Matrix.vecMulstatement and proof · cited by 148
- Matrix.colproof · cited by 56
Cited by2
Results whose statement or proof uses this declaration.
- Matrix.SeparatingLeft.eq_zero_of_vecMul_eq_zeroproof · cited by 2
- Matrix.nondegenerate_iff_forall_vecMul_and_mulVec_eq_zeroproof · cited by 0