VectorSpaces.hs

{-# LANGUAGE NoImplicitPrelude #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE ExistentialQuantification #-}
{-# LANGUAGE Rank2Types #-}
{-# LANGUAGE AllowAmbiguousTypes #-}

module VectorSpaces where

import Groups
import Categories
import Numbers

class (AbelianGroup v, Field f) => VectorSpace v f where
    infixl 7 *^
    (*^) :: f -> v -> v
    (*^) a b = b ^* a
    infixl 7 ^*
    (^*) :: v -> f -> v
    (^*) a b = b *^ a

instance (Field f) => VectorSpace f f where
    (*^) a b = a * b

class (Floating f, VectorSpace v f) => InnerProductSpace v f where
    dot :: v -> v -> f

instance (Floating f) => InnerProductSpace f f where
    dot = (*)
{-
data LinearMap v1 v2 = LinearMap { getMap :: forall f1 f2 . (VectorSpace v1 f1, VectorSpace v2 f2) => v1 -> v2 }
instance Category LinearMap VectorSpace where
    f . g = LinearMap { getMap = (getMap f . getMap g) }
-}