Theorems · Theorem · linear algebra

Module.Basis.orientation_comp_linearEquiv_eq_neg_iff_det_neg

∀ {R : Type u_1} [inst : CommRing R] [inst_1 : LinearOrder R] [inst_2 : IsStrictOrderedRing R] {M : Type u_2}
  [inst_3 : AddCommGroup M] [inst_4 : Module R M] {ι : Type u_3} [inst_5 : Fintype ι] [inst_6 : DecidableEq ι]
  (e : Module.Basis ι R M) (f : M ≃ₗ[R] M), (e.map f).orientation = -e.orientation ↔ LinearMap.det ↑f < 0

Composing a basis with a linear equiv gives the negation of that orientation if and only if the determinant is negative.

Defined in

Mathlib.LinearAlgebra.Orientation

Cited by

0 results in Mathlib

Foundations

Depth 132 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLinearOrderIsStrictOrderedRingAddCommGroupModuleFintypeDecidableEq