Theorems · Theorem · commutative algebra

Module.Basis.traceDual_eq_iff

∀ {K : Type u_4} {L : Type u_5} [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L] {ι : Type w}
  [inst_3 : FiniteDimensional K L] [inst_4 : Algebra.IsSeparable K L] [inst_5 : Finite ι] [inst_6 : DecidableEq ι]
  {b : Module.Basis ι K L} {v : ι → L},
  ⇑b.traceDual = v ↔ ∀ (i j : ι), ((Algebra.traceForm K L) (v i)) (b j) = if j = i then 1 else 0

A family of vectors v is the dual for the trace of the basis b if and only if ∀ i j, Tr(v i * b j) = δ_ij.

Defined in

Mathlib.RingTheory.Trace.Basic

Cited by

2 results in Mathlib

Foundations

Depth 184 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldFieldAlgebraFiniteDimensionalAlgebra.IsSeparableFiniteDecidableEq