Theorems · Theorem · linear algebra

TensorProduct.equivFinsuppOfBasisRight_apply

∀ {R : Type u_1} {M : Type u_3} {N : Type u_4} {κ : Type u_6} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]
  [inst_2 : Module R M] [inst_3 : AddCommMonoid N] [inst_4 : Module R N] [inst_5 : DecidableEq κ]
  (𝒞 : Module.Basis κ R N) (x : TensorProduct R M N) (i : κ),
  ((TensorProduct.equivFinsuppOfBasisRight 𝒞) x) i = (TensorProduct.rid R M) ((LinearMap.lTensor M (𝒞.coord i)) x)

Given a basis 𝒞 of N, x ∈ M ⊗ N can be written as ∑ᵢ mᵢ ⊗ 𝒞 i. The coefficient mᵢ equals the i-th coordinate functional applied to the right tensor factor.

Defined in

Mathlib.LinearAlgebra.TensorProduct.Basis

Cited by

0 results in Mathlib

Foundations

Depth 98 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringAddCommMonoidModuleAddCommMonoidModuleDecidableEq