Theorems · Theorem · linear algebra

Matrix.Nonsingular.linearIndependent_col

∀ {R : Type u_1} {n : Type u_3} [inst : CommSemiring R] [inst_1 : Fintype n] [inst_2 : DecidableEq n] {A : Matrix n n R}
  [IsCancelAdd R], A.Nonsingular → LinearIndependent R A.col

If a matrix over a commutative semiring with cancellative addition is nonsingular, then its columns are linearly independent. Generalizes Matrix.linearIndependent_cols_of_det_ne_zero.

Defined in

Mathlib.LinearAlgebra.Matrix.Nonsingular

Cited by

2 results in Mathlib

Foundations

Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringFintypeDecidableEqIsCancelAdd