module Control.Algebra.VectorSpace
import Control.Algebra
infixl 5 <#>
infixr 2 <||>
||| A module over a ring is an additive abelian group of 'vectors' endowed with a
||| scale operation multiplying vectors by ring elements, and distributivity laws
||| relating the scale operation to both ring addition and module addition.
||| Must satisfy the following laws:
|||
||| + Compatibility of scalar multiplication with ring multiplication:
||| forall a b v, a <#> (b <#> v) = (a <.> b) <#> v
||| + Ring unity is the identity element of scalar multiplication:
||| forall v, unity <#> v = v
||| + Distributivity of `<#>` and `<+>`:
||| forall a v w, a <#> (v <+> w) == (a <#> v) <+> (a <#> w)
||| forall a b v, (a <+> b) <#> v == (a <#> v) <+> (b <#> v)
class (RingWithUnity a, AbelianGroup b) => Module a b where
(<#>) : a -> b -> b
||| A vector space is a module over a ring that is also a field
class (Field a, Module a b) => VectorSpace a b where {}
||| An inner product space is a module – or vector space – over a ring, with a binary function
||| associating a ring value to each pair of vectors.
class Module a b => InnerProductSpace a b where
(<||>) : b -> b -> a