Theorems · Theorem · geometry

AffineBasis.det_smul_coords_eq_cramer_coords

∀ {ι : Type u₁} {k : Type u₂} {V : Type u₃} {P : Type u₄} [inst : AddCommGroup V] [inst_1 : AddTorsor V P]
  [inst_2 : CommRing k] [inst_3 : Module k V] [inst_4 : DecidableEq ι] [inst_5 : Fintype ι] (b b₂ : AffineBasis ι k P)
  (x : P), (b.toMatrix ⇑b₂).det • b₂.coords x = (b.toMatrix ⇑b₂).transpose.cramer (b.coords x)

If we fix a background affine basis b, then for any other basis b₂, we can characterise the barycentric coordinates provided by b₂ in terms of determinants relative to b.

Defined in

Mathlib.LinearAlgebra.AffineSpace.Matrix

Cited by

0 results in Mathlib

Foundations

Depth 106 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddCommGroupAddTorsorCommRingModuleDecidableEqFintype