Theorems · Theorem · linear algebra
Matrix.mulVec_injective_iff
∀ {R : Type u_1} [inst : CommSemiring R] {m : Type u_4} {n : Type u_5} [inst_1 : Fintype n] {M : Matrix m n R},
Function.Injective M.mulVec ↔ LinearIndependent R M.col- Defined in
- Mathlib.LinearAlgebra.Matrix.ToLin
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringFintype
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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
- LinearIndependentstatement and proof · cited by 560
- Matrix.mulVecstatement · cited by 267
- Matrix.colstatement and proof · cited by 56
- Matrix.vecMul_transposeproof · cited by 8
Cited by5
Results whose statement or proof uses this declaration.
- Matrix.isLeftRegular_iff_nonsingularproof · cited by 3
- Matrix.linearIndependent_rows_iff_isUnitproof · cited by 2
- Matrix.Nonsingular.linearIndependent_colproof · cited by 2
- Matrix.linearIndependent_cols_of_isUnitproof · cited by 0
- Matrix.Nonsingular.of_leftRegularproof · cited by 0