Math.Algebras.VectorSpace

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

Documentation

newtype Vect k b

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

Fractional k => Fractional (LaurentPoly k)

Num k => Monad (Vect k)

Functor (Vect k)

HopfAlgebra (LaurentPoly Q) (SL2q String)

Bialgebra (LaurentPoly Q) (SL2q String)

Bialgebra (LaurentPoly Q) (M2q String)

Coalgebra (LaurentPoly Q) (SL2q String)

Coalgebra (LaurentPoly Q) (M2q String)

Algebra (LaurentPoly Q) (SL2q String)

Algebra (LaurentPoly Q) (M2q String)

Algebra (LaurentPoly Q) (Aq02 String)

Algebra (LaurentPoly Q) (Aq20 String)

Comodule (LaurentPoly Q) (M2q String) (Aq20 String)

(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 b

The zero vector

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

Addition of vectors

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

Addition of vectors (same as add)

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

Negation of vector

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

Scalar multiplication (on the left)

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

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

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

Scalar multiplication on the right

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

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

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

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 b

newtype EBasis

Constructors

E Int

Instances

Eq EBasis

Ord EBasis

Show EBasis

Num k => Module k Mat2 EBasis