Updates:
Status: Accepted
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Comment #1 on issue 2301 by asmeurer: RFC: Basis, and/or VectorSpace class
required
http://code.google.com/p/sympy/issues/detail?id=2301
Sounds like a good idea to me.
At my university, we have a sophomore level linear algebra class, which
focuses on matrices and calculation, and a senior level class that focuses
on the theory and proving the results (I am in this senior class right now
actually). The senior class proves everything in terms of general vector
spaces whenever possible, only falling back to matrices when necessary for
calculation. It think that that could make a good model in SymPy too.
Then you could, for example, consider linear transformations on the space
of polynomials P(R) without having to compute the matrix of the
transformation explicitly (this will be especially nice when the matrix is
nontrivial, such as when you want to consider it over some orthonormal
basis). You could also, in limited cases, consider things over infinite
dimensional vector spaces, though this is more advanced, harder to compute,
and probably less useful in general, but anyway the framework should allow
for it.
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