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Mathlib.LinearAlgebra.FreeModule.Finite.Matrix

Finite and free modules using matrices #

We provide some instances for finite and free modules involving matrices.

Main results #

Module.Free.linearMap : if M and N are finite and free, then M →ₗ[R] N is free.
Module.Finite.ofBasis : A free module with a basis indexed by a Fintype is finite.
Module.Finite.linearMap : if M and N are finite and free, then M →ₗ[R] N is finite.

instance Module.Free.linearMap (R : Type u) (M : Type v) (N : Type w) [CommRing R] [AddCommGroup M] [Module R M] [Module.Free R M] [AddCommGroup N] [Module R N] [Module.Free R N] [Module.Finite R M] [Module.Finite R N] :

Module.Free R (M →ₗ[R] N)

Equations

Module.Free.linearMap = Module.Free.linearMap.proof_1

instance Module.Finite.linearMap {R : Type u} (M : Type v) (N : Type w) [CommRing R] [AddCommGroup M] [Module R M] [Module.Free R M] [AddCommGroup N] [Module R N] [Module.Free R N] [Module.Finite R M] [Module.Finite R N] :

Module.Finite R (M →ₗ[R] N)

Equations

@Module.Finite.linearMap = @Module.Finite.linearMap.proof_1

instance Module.Finite.addMonoidHom (M : Type v) (N : Type w) [AddCommGroup M] [Module.Finite ℤ M] [Module.Free ℤ M] [AddCommGroup N] [Module.Finite ℤ N] [Module.Free ℤ N] :

Module.Finite ℤ (M →+ N)

Equations

Module.Finite.addMonoidHom = Module.Finite.addMonoidHom.proof_1

instance Module.Free.addMonoidHom (M : Type v) (N : Type w) [AddCommGroup M] [Module.Finite ℤ M] [Module.Free ℤ M] [AddCommGroup N] [Module.Finite ℤ N] [Module.Free ℤ N] :

Module.Free ℤ (M →+ N)

Equations

Module.Free.addMonoidHom = Module.Free.addMonoidHom.proof_1

theorem FiniteDimensional.finrank_linearMap (R : Type u) (M : Type v) (N : Type w) [CommRing R] [StrongRankCondition R] [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] [AddCommGroup N] [Module R N] [Module.Free R N] [Module.Finite R N] :

FiniteDimensional.finrank R (M →ₗ[R] N) = FiniteDimensional.finrank R M * FiniteDimensional.finrank R N

The finrank of M →ₗ[R] N is (finrank R M) * (finrank R N).

theorem Matrix.rank_vecMulVec {K : Type u} {m : Type u} {n : Type u} [CommRing K] [StrongRankCondition K] [Fintype n] [DecidableEq n] (w : m → K) (v : n → K) :

LinearMap.rank (↑Matrix.toLin' (Matrix.vecMulVec w v)) ≤ 1