"Abstract" linear algebra

asked 2010-09-07 01:28:29 +0200

marco
83 ●4 ●5 ●16

updated 2023-05-19 22:12:18 +0200

5125 ●3 ●43 ●111

This is more of a general question about whether we can do with Sage what we normally do as mathematicians on paper and in our minds. For example, if I do

V = VectorSpace(QQ,4)

W = VectorSpace(QQ,4)

V==W

I get: True. This is quite disturbing - while V and W are isomorphic they should not be identical. As I understand it, a vectorspace over QQ for Sage is just QQ^4, that's it. In other words, the Linear algebra package, while excellent, is not really about vector spaces but rather about arrays of numbers.

Any thoughts on this desire of mine to have a genuine "coordinate-free" approach?

answered 2010-09-07 11:31:05 +0200

niles
4519 ●23 ●72 ●132 http://nilesjohnson.net/

I like coordinate-free linear algebra, but your issue here may simply be that == is not the "is identical" operator. Consider the following examples:

sage: 3 == Mod(3,7)
True
sage: 3 == Mod(3,5)
True
sage: Mod(3,5) == Mod(3,7)
False
sage: Q = PolynomialRing(QQ,'t')
sage: 3 == Q(3)
True
sage: 3 in Q
True
sage: R = PolynomialRing(RR,'w,v')
sage: 3 == R(3)
True
sage: Q(3) == R(3)
False

Given this behavior, I don't find it particularly surprising that V == W returns True above!

Of course, id(V) == id(W) also returns True, which is a little more surprising (to me). And the free module documentation includes the following note:

Note: In Sage it is the case that there is only one dense and one sparse free ambient module of rank n over R.

So it seems it's not even _possible_ to make two separate copies of a free module with a given rank. That seems a little limiting, which is your original point, I guess :) Maybe it's more accurate to think of VectorSpace as short for "VectorSpaceWithCanonicalBasis".

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Comments

Thanks for the comments! Given that they went to the trouble of implementing categories, this seems particularly strange. But I think this "problem" is common to all CAS.There must be a reference out there which works out the fundamentals for a CAS with higher fidelity.

marco ( 2010-09-07 12:41:48 +0200 )edit

This behavior of "VectorSpace" -- which is *just* shorthand for K^n -- is very much by design. It's the same as what Magma does. Also, all this " trouble of implementing categories" was done 4 years after VectorSpace was implemented. The combinatorics code maybe has other more general modules.

William Stein ( 2010-09-08 11:32:40 +0200 )edit

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