Documentation

R-modules. A module v over a ring Groundring v is an abelian group with a linear multiplication. The hat ^ denotes the side of an operation on which the vector stands, i.e. a *^ v for v a vector.

A minimal definition should include the type Groundring and the implementations of zeroVector, ^+^, and one of *^ or ^*.

The following laws must be satisfied:

• v1 ^+^ v2 == v2 ^+^ v1

• a *^ zeroVector == zeroVector

• @a *^ (v1 ^+^ v2) == a *^ v1 ^+^ a*^ v2

• a *^ v == v ^* a

• negateVector v == (-1) *^ v

• v1 ^-^ v2 == v1 ^+^ negateVector v2

A vector space is a module over a field, i.e. a commutative ring with inverses.

It needs to satisfy the axiom v ^ a == (1a) *^ v, which is the default implementation.

The ground ring of a vector space is required to be commutative and to possess inverses. It is then called the "ground field". Commutativity amounts to the law a * b = b * a, and the existence of inverses is given by the requirement of the Fractional type class.

An inner product space is a module with an inner product, i.e. a map dot satisfying

• v1 dot v2 == v2 dot v1

• (v1 ^+^ v2) dot v3 == v1 dot v3 ^+^ v2 dot v3

• (a *^ v1) dot v2 == a *^ v1 dot v2

A normed space is a module with a norm, i.e. a function norm satisfying

• norm (a ^* v) = a ^* norm v

• norm (v1 ^+^ v2) <= norm v1 ^+^ norm v2 (the "triangle inequality")

A typical example is sqrt (v dot v), for an inner product space.