Theorems · Definition · linear algebra

LinearMap.ker

{R : Type u_1} →
  {R₂ : Type u_2} →
    {M : Type u_5} →
      {M₂ : Type u_7} →
        [inst : Semiring R] →
          [inst_1 : Semiring R₂] →
            [inst_2 : AddCommMonoid M] →
              [inst_3 : AddCommMonoid M₂] →
                [inst_4 : Module R M] → [inst_5 : Module R₂ M₂] → {τ₁₂ : R →+* R₂} → (M →ₛₗ[τ₁₂] M₂) → Submodule R M

The kernel of a linear map f : M → M₂ is defined to be comap f ⊥. This is equivalent to the set of x : M such that f x = 0. The kernel is a submodule of M.

Defined in

Mathlib.Algebra.Module.Submodule.Ker

Cited by

848 results in Mathlib

Foundations

Depth 25 from the axioms, rests on 228 definitions · uses propext, Quot.sound

Assumes

SemiringSemiringAddCommMonoidAddCommMonoidModuleModule