Theorems · Theorem · linear algebra

LinearMap.trace_eq_contract_of_basis

∀ {R : Type u_1} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : Module R M] {ι : Type u_5}
  [Finite ι] (b : Module.Basis ι R M), LinearMap.trace R M ∘ₗ dualTensorHom R M M = contractLeft R M

The trace of a linear map corresponds to the contraction pairing under the isomorphism End(M) ≃ M* ⊗ M

Defined in

Mathlib.LinearAlgebra.Trace

Cited by

2 results in Mathlib

Foundations

Depth 100 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleFinite