HaskellForMaths-0.3.1: Combinatorics, group theory, commutative algebra, non-commutative algebra

Math.Algebras.VectorSpace

Description

A module defining the type and operations of free k-vector spaces over a basis b (for a field k)

Synopsis

# Documentation

data Vect k b Source

Given a field type k (ie a Fractional instance), Vect k b is the type of the free k-vector space over the basis type b. Elements of Vect k b consist of k-linear combinations of elements of b.

Constructors

 V [(b, k)]

Instances

 Num k => Monad (Vect k) Functor (Vect k) (Eq k, Eq b) => Eq (Vect k b) (Num k, Eq b, Ord b, Show b, Algebra k b) => Num (Vect k b) (Ord k, Ord b) => Ord (Vect k b) (Num k, Show b) => Show (Vect k b)

zero :: Vect k bSource

The zero vector

add :: (Ord b, Num k) => Vect k b -> Vect k b -> Vect k bSource

(<+>) :: (Ord b, Num k) => Vect k b -> Vect k b -> Vect k bSource

neg :: Num k => Vect k b -> Vect k bSource

Negation of vector

smultL :: Num k => k -> Vect k b -> Vect k bSource

Scalar multiplication (on the left)

(*>) :: Num k => k -> Vect k b -> Vect k bSource

Same as smultL. Mnemonic is multiply through (from the left)

smultR :: Num k => Vect k b -> k -> Vect k bSource

Scalar multiplication on the right

(<*) :: Num k => Vect k b -> k -> Vect k bSource

Same as smultR. Mnemonic is multiply through (from the right)

nf :: (Ord b, Num k) => Vect k b -> Vect k bSource

Convert an element of Vect k b into normal form. Normal form consists in having the basis elements in ascending order, with no duplicates, and all coefficients non-zero

linear :: (Ord b, Num k) => (a -> Vect k b) -> Vect k a -> Vect k bSource

newtype EBasis Source

Constructors

 E Int

Instances

 Eq EBasis Ord EBasis Show EBasis Num k => Module k Mat2 EBasis