Theorems · Theorem · linear algebra

LinearMap.BilinForm.not_linearIndependent_of_apply_mul_apply_eq

∀ {R : Type u_1} {M : Type u_5} [inst : CommRing R] [inst_1 : LinearOrder R] [IsStrictOrderedRing R]
  [inst_3 : AddCommGroup M] [inst_4 : Module R M] (B : LinearMap.BilinForm R M),
  (∀ (x : M), x ≠ 0 → 0 < (B x) x) → ∀ (x y : M), (B x) y * (B y) x = (B x) x * (B y) y → ¬LinearIndependent R ![x, y]

The equality case of Cauchy-Schwarz.

Defined in

Mathlib.LinearAlgebra.SesquilinearForm.Basic

Cited by

1 results in Mathlib

Foundations

Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLinearOrderIsStrictOrderedRingAddCommGroupModule