Theorems · Definition · linear algebra

Submodule.orthogonalBilin

{R : Type u_1} →
  {R₁ : Type u_2} →
    {R₂ : Type u_3} →
      {M : Type u_4} →
        {M₁ : Type u_5} →
          {M₂ : Type u_6} →
            [inst : CommSemiring R] →
              [inst_1 : CommSemiring R₁] →
                [inst_2 : CommSemiring R₂] →
                  [inst_3 : AddCommMonoid M] →
                    [inst_4 : Module R M] →
                      [inst_5 : AddCommMonoid M₁] →
                        [inst_6 : Module R₁ M₁] →
                          [inst_7 : AddCommMonoid M₂] →
                            [inst_8 : Module R₂ M₂] →
                              {I₁ : R₁ →+* R} →
                                {I₂ : R₂ →+* R} → (M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M) → Submodule R₁ M₁ → Submodule R₂ M₂

The orthogonal complement of a submodule N with respect to some bilinear map is the set of elements x which are orthogonal to all elements of N; i.e., for all y in N, B x y = 0. Note that for general (neither symmetric nor antisymmetric) bilinear maps this definition has a chirality; in addition to this "left" orthogonal complement one could define a "right" orthogonal complement for which, for all y in N, B y x = 0. This variant definition is not currently provided in mathlib.

Defined in

Mathlib.LinearAlgebra.SesquilinearForm.Orthogonal

Cited by

10 results in Mathlib

Foundations

Depth 33 from the axioms · uses propext, Quot.sound

Assumes

CommSemiringCommSemiringCommSemiringAddCommMonoidModuleAddCommMonoidModuleAddCommMonoidModule