Hi!

This README file describes the basics of the VectorSpace project and how it works.  So, I hope you're reading this before checking out the code!

Note: This project is written entirely in C++, mainly to take advantage of the operator overloading this language allows.  However, C++ also has its limitations, as noticeable in the painful methods used to concatenate strings with various primitives.

The idea:

The VectorSpaceCheck template class takes in a space class and its dimension.  Once this has been set up, any (or all) or the 8 vector space axioms can be checked for this space.
Ideally, this would also be able to check the 2 closure axioms, however this has not been implemented yet.

So, to work with this, I have included a basic n-dimensional vector space template, V (can specify element type, scalar multiple type, and dimension), where addition and scalar multiplication are defined in the standard element-wise form.  There is also a 2D vector space, W2 (with V as a base class) which redefined addition and scalar multiplication in a non-standard way.  Note that this space satisfies all the axioms.

The functionality:

To get this to work, I wrote the SymbCheck_Real class - the idea of which is to take in a string representation of an expression and parse it into 2 vectors (one vector of all the variables contained in the expression, and one vector of doubles with the corresponding coefficients).  Then, using this class, basic algebra can be performed.

To actually check the axioms, initialize an instantiation of VectorSpaceCheck with a space class of type SymbCheck_Real - then, the axioms can be checked algebraically to see if they are actually true for all vectors.
For example, to check W2...
	VectorSpaceCheck<W2<SymbCheck_Real, SymbCheck_Real>, 2> CHECK;
Now, the variable CHECK can be used to check the axioms.  For example:
	bool ax1 = CHECK.checkAxiom1();
There are some examples of this use in main.cpp.

All code (especially SymbCheck_Real) is thoroughly commented!  Details of function implementation is all contained in the project files.

Why did I do this?

... Well, I thought it was cool! :)

Disclaimers:

Many of the functions (particulary in SymbCheck_Real) only work in conjuction with each other, and used in a specific order.  Beware of switching the order they are called in, especially in the constructors!

Also, just a heads-up that there is not much error-checking in terms of arguments for many of the functions.

Coming up:

1. Implement checking for the closure axioms!
2. Write SymbCheck_Complex class (SymbCheck_Real only works for real-number algebra)
3. Get this to work with matrix vector spaces
4. Maybe find more examples of vector spaces which define scalar multiplication/addition in non-standard ways; and, find counter-examples i.e. spaces which are not vector spaces (axioms fail)
5. Far future: ideally, this would work with function spaces too (probably implementing for polynomials would be the first step here)

Well, I hope you enjoyed my vector space project!  Feel free to fork the repository and mess around with it however you want.  If you have any improvements/suggestions to make please let me know!


Thanks! :)