VasCalc Documentation - vectorSpace

The vectorSpace package is a small package used to set up quotients of finite dimensional vector spaces, with an interface through Mathematica. By default, vectorSpace works over arbitrarily-sized rationals, using the BigRational implementation written by Eric Laroche (see VasCalc - Bibliography). See below for instructions on using other fields. The main advantage in using this package is that by being less general, it stands to perform much faster than similar functions in Mathematica.

Installation

If you wish to use this package on its own, follow the directions below.

Create a directory on your file system named vectorSpace, and download the following four files from the repository into it:

• Coefficient.class,
• QuotientSpace.class,
• AlgebraVector.class,
• BigRational.class.

Also, download vectorSpace.m (the Mathematica interface) to any location.

Now open vectorSpace.m in a text editor. Near the top, there is a line that reads

AddToClassPath["some_path"]

Change some_path to the location in which the directory vectorSpace resides (ie. the parent directory of vectorSpace). Make sure you keep the quotes around the path name.

Usage

In Mathematica, type

In[ ]:= << /path_to_/vectorSpace.m

to load the definitions.

We need to start by setting up a finite-dim vector space to work with. The command NewVS[n] returns a reference to a Java instance of the class QuotientSpace. This class represents a v.s. of dimension n, along with any relations between the basis vectors (described below). Here we create a vector space of dimension 5:

In[ ] := qs = NewVS[5];

In the vector space package, vectors are stored as lists of ordered pairs {index, coeff}, where the index refers to a fixed implicit original basis. For convenience, the Mathematica interface allows one to label the basis elements with Mathematica symbols by passing a Basis[..] object to the operations described below, and hence you can use all of Mathematica's symbolic manipulations on these expressions. Note, however, that this labelling does not change the internal representation of the vectors; if you change the basis you use, it's unlikely you'll get meaningful results, as you may have already defined relations in terms of another basis.

To set up an initial basis, say of elements e0, ... , e4:

In[] := b= Basis[e0,e1,e2,e3,e4];

Here are the operations one can perform:

• To introduce a relation, use ModBy[v, quotientspace, basis], which introduces the relation v=0 (where v is written in terms of basis) in the given vector space quotientspace.

• The command GetDimension[quotientspace], returns the dimension of the quotient (that is, taking into account the relations defined).

• The command ReduceQuotient[v, quotientspace, basis] returns a vector equivalent to v modulo the relations defined in quotientspace. Both v and the output are written as linear combinations in terms of the given basis.

• If you need it, the command ResolveVector[v, basis] converts a vector as a formal sum into a list of {index, coeff} pairs.

Here are some examples, using the earlier setup: Let's introduce a relation

In[ ] := ModBy[3e0 + e3 - 2e4, qs, b]

Now the quotient space has one less dimension:

In[ ] := GetDimension[qs]

Out[ ] = 4

In[ ] := ResolveQuotient[e0 + (1/3)e1 - e4, qs, b]

Out[ ] = e1/4 - (2*e3)/3 + e4/3

In[ ]:= ResolveVector[3e1 +2/3 e0 - 2 e2, b]

Out[ ] = { {0,2/3}, {1,3}, {2,-2} }