Scalar multiplication of vectors on rings

Content created by Egbert Rijke, Fredrik Bakke and Jonathan Prieto-Cubides.

Created on 2023-02-20.
Last modified on 2023-11-24.

module linear-algebra.scalar-multiplication-vectors-on-rings where
Imports 
open import elementary-number-theory.natural-numbers

open import foundation.action-on-identifications-binary-functions
open import foundation.dependent-pair-types
open import foundation.identity-types
open import foundation.universe-levels

open import group-theory.endomorphism-rings-abelian-groups
open import group-theory.homomorphisms-abelian-groups

open import linear-algebra.vectors
open import linear-algebra.vectors-on-rings

open import ring-theory.homomorphisms-rings
open import ring-theory.modules-rings
open import ring-theory.rings

Definition

Scalar multiplication of vectors on rings

module _
  {l : Level} (R : Ring l)
  where

  scalar-mul-vec-Ring : {n : ℕ} (r : type-Ring R) → vec-Ring R n → vec-Ring R n
  scalar-mul-vec-Ring r empty-vec = empty-vec
  scalar-mul-vec-Ring r (x ∷ v) = mul-Ring R r x ∷ scalar-mul-vec-Ring r v

  associative-scalar-mul-vec-Ring :
    {n : ℕ} (r s : type-Ring R) (v : vec-Ring R n) →
    scalar-mul-vec-Ring (mul-Ring R r s) v ＝
    scalar-mul-vec-Ring r (scalar-mul-vec-Ring s v)
  associative-scalar-mul-vec-Ring r s empty-vec = refl
  associative-scalar-mul-vec-Ring r s (x ∷ v) =
    ap-binary _∷_
      ( associative-mul-Ring R r s x)
      ( associative-scalar-mul-vec-Ring r s v)

  unit-law-scalar-mul-vec-Ring :
    {n : ℕ} (v : vec-Ring R n) → scalar-mul-vec-Ring (one-Ring R) v ＝ v
  unit-law-scalar-mul-vec-Ring empty-vec = refl
  unit-law-scalar-mul-vec-Ring (x ∷ v) =
    ap-binary _∷_ (left-unit-law-mul-Ring R x) (unit-law-scalar-mul-vec-Ring v)

  left-distributive-scalar-mul-add-vec-Ring :
    {n : ℕ} (r : type-Ring R) (v1 v2 : vec-Ring R n) →
    scalar-mul-vec-Ring r (add-vec-Ring R v1 v2) ＝
    add-vec-Ring R (scalar-mul-vec-Ring r v1) (scalar-mul-vec-Ring r v2)
  left-distributive-scalar-mul-add-vec-Ring r empty-vec empty-vec = refl
  left-distributive-scalar-mul-add-vec-Ring r (x ∷ v1) (y ∷ v2) =
    ap-binary _∷_
      ( left-distributive-mul-add-Ring R r x y)
      ( left-distributive-scalar-mul-add-vec-Ring r v1 v2)

  right-distributive-scalar-mul-add-vec-Ring :
    {n : ℕ} (r s : type-Ring R) (v : vec-Ring R n) →
    scalar-mul-vec-Ring (add-Ring R r s) v ＝
    add-vec-Ring R (scalar-mul-vec-Ring r v) (scalar-mul-vec-Ring s v)
  right-distributive-scalar-mul-add-vec-Ring r s empty-vec = refl
  right-distributive-scalar-mul-add-vec-Ring r s (x ∷ v) =
    ap-binary _∷_
      ( right-distributive-mul-add-Ring R r s x)
      ( right-distributive-scalar-mul-add-vec-Ring r s v)

Properties

Scalar multiplication defines an Ab-endomorphism of vec-Rings, and this mapping is a ring homomorphism R → End(vec R n)

  scalar-mul-vec-Ring-endomorphism :
    (n : ℕ) (r : type-Ring R) → hom-Ab (vec-Ring-Ab R n) (vec-Ring-Ab R n)
  pr1 (scalar-mul-vec-Ring-endomorphism n r) = scalar-mul-vec-Ring r
  pr2 (scalar-mul-vec-Ring-endomorphism n r) {x} {y} =
    left-distributive-scalar-mul-add-vec-Ring r x y

  scalar-mul-hom-Ring :
    (n : ℕ) → hom-Ring R (endomorphism-ring-Ab (vec-Ring-Ab R n))
  pr1 (pr1 (scalar-mul-hom-Ring n)) = scalar-mul-vec-Ring-endomorphism n
  pr2 (pr1 (scalar-mul-hom-Ring n)) {k1} {k2} =
    eq-htpy-hom-Ab
      ( vec-Ring-Ab R n)
      ( vec-Ring-Ab R n)
      ( right-distributive-scalar-mul-add-vec-Ring k1 k2)
  pr1 (pr2 (scalar-mul-hom-Ring n)) {k1} {k2} =
    eq-htpy-hom-Ab
      ( vec-Ring-Ab R n)
      ( vec-Ring-Ab R n)
      ( associative-scalar-mul-vec-Ring k1 k2)
  pr2 (pr2 (scalar-mul-hom-Ring n)) =
    eq-htpy-hom-Ab
      ( vec-Ring-Ab R n)
      ( vec-Ring-Ab R n)
      ( unit-law-scalar-mul-vec-Ring)

  vec-left-module-Ring : (n : ℕ) → left-module-Ring l R
  vec-left-module-Ring n = vec-Ring-Ab R n , scalar-mul-hom-Ring n

Recent changes