Theorems · Definition · linear algebra

LinearMap.toMatrixAlgEquiv

{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 → (M₁ →ₗ[R] M₁) ≃ₐ[R] Matrix n n R

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

Defined in

Mathlib.LinearAlgebra.Matrix.ToLin

Cited by

19 results in Mathlib

Foundations

Depth 92 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringFintypeDecidableEqAddCommMonoidModule