Theorems · Definition · linear algebra

Module.Basis.constr

{M' : Type u_7} →
  [inst : AddCommMonoid M'] →
    {ι : Type u_10} →
      {R : Type u_11} →
        {M : Type u_12} →
          [inst_1 : Semiring R] →
            [inst_2 : AddCommMonoid M] →
              [inst_3 : Module R M] →
                Module.Basis ι R M →
                  [inst_4 : Module R M'] →
                    (S : Type u_13) →
                      [inst_5 : Semiring S] →
                        [inst_6 : Module S M'] → [inst_7 : SMulCommClass R S M'] → (ι → M') ≃ₗ[S] M →ₗ[R] M'

Construct a linear map given the value at the basis, called Basis.constr b S f where b is a basis, f is the value of the linear map over the elements of the basis, and S is an extra semiring (typically S = R or S = ℕ). This definition is parameterized over an extra Semiring S, such that SMulCommClass R S M' holds. If R is commutative, you can set S := R; if R is not commutative, you can recover an AddEquiv by setting S := ℕ. See library note [bundled maps over different rings].

Defined in

Mathlib.LinearAlgebra.Basis.Defs

Cited by

37 results in Mathlib

Foundations

Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddCommMonoidSemiringAddCommMonoidModuleModuleSemiringModuleSMulCommClass