Theorems · Theorem · linear algebra

Orientation.map_eq_det_inv_smul

∀ {R : Type u_1} [inst : Field 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] [FiniteDimensional R M]
  (x : Orientation R M ι) (f : M ≃ₗ[R] M),
  Fintype.card ι = Module.finrank R M → (Orientation.map ι f) x = (LinearEquiv.det f)⁻¹ • x

The value of Orientation.map when the index type has cardinality equal to the finite dimension, in terms of f.det.

Defined in

Mathlib.LinearAlgebra.Orientation

Cited by

2 results in Mathlib

Foundations

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

Assumes

FieldLinearOrderIsStrictOrderedRingAddCommGroupModuleFintypeFiniteDimensional