Theorems · Definition · linear algebra

Matrix.toLinAlgEquiv

{R : Type u_1} →
  [inst : CommSemiring R] →
    {n : Type u_4} →
      [inst_1 : Fintype n] →
        [inst_2 : DecidableEq n] →
          {M₁ : Type u_5} →
            [inst_3 : AddCommMonoid M₁] → [inst_4 : Module R M₁] → Module.Basis n R M₁ → Matrix n n R ≃ₐ[R] M₁ →ₗ[R] M₁

Given a basis of a module M₁ over a commutative ring R, we get an algebra equivalence between square matrices over R indexed by the basis and linear maps M₁ →ₗ M₁.

Defined in

Mathlib.LinearAlgebra.Matrix.ToLin

Cited by

13 results in Mathlib

Foundations

Depth 93 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringFintypeDecidableEqAddCommMonoidModule