# Bases and dimensionality of tensor products of modules #

These can not go into LinearAlgebra.TensorProduct since they depend on LinearAlgebra.FinsuppVectorSpace which in turn imports LinearAlgebra.TensorProduct.

def Basis.tensorProduct {R : Type u_1} {S : Type u_2} {M : Type u_3} {N : Type u_4} {ι : Type u_5} {κ : Type u_6} [] [] [Algebra R S] [] [Module R M] [Module S M] [] [] [Module R N] (b : Basis ι S M) (c : Basis κ R N) :
Basis (ι × κ) S ()

If b : ι → M and c : κ → N are bases then so is fun i ↦ b i.1 ⊗ₜ c i.2 : ι × κ → M ⊗ N.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem Basis.tensorProduct_apply {R : Type u_1} {M : Type u_3} {N : Type u_4} {ι : Type u_5} {κ : Type u_6} [] [] [Module R M] [] [Module R N] (b : Basis ι R M) (c : Basis κ R N) (i : ι) (j : κ) :
(b.tensorProduct c) (i, j) = b i ⊗ₜ[R] c j
theorem Basis.tensorProduct_apply' {R : Type u_1} {M : Type u_3} {N : Type u_4} {ι : Type u_5} {κ : Type u_6} [] [] [Module R M] [] [Module R N] (b : Basis ι R M) (c : Basis κ R N) (i : ι × κ) :
(b.tensorProduct c) i = b i.1 ⊗ₜ[R] c i.2
@[simp]
theorem Basis.tensorProduct_repr_tmul_apply {R : Type u_1} {M : Type u_3} {N : Type u_4} {ι : Type u_5} {κ : Type u_6} [] [] [Module R M] [] [Module R N] (b : Basis ι R M) (c : Basis κ R N) (m : M) (n : N) (i : ι) (j : κ) :
((b.tensorProduct c).repr (m ⊗ₜ[R] n)) (i, j) = (b.repr m) i * (c.repr n) j