basis_matrix returns matrix over wrong ring

[ncalexan]

sage: (ZZ**3).basis_matrix()

[1 0 0]
[0 1 0]
[0 0 1]
sage: (ZZ**3).basis_matrix().parent()
Full MatrixSpace of 3 by 3 dense matrices over Rational Field

That should be over the integer ring; you can't do (ZZ**3).basis_matrix().smith_form() otherwise.

[was]

This is not a bug, it is a design decision. ZZ modules in Sage in general need *not* be embedded in ZZ^{n -- they can be in QQ}n. For consistency the basis matrix is thus always over QQ. Here's an example of a module over ZZ but for which the basis matrix is over QQ.

sage: A = (ZZ^3).span([[1,2,3], [1/3,5,6]])
sage: A
Free module of degree 3 and rank 2 over Integer Ring
Echelon basis matrix:
[1/3   5   6]
[  0  13  15]
sage: A.basis_matrix()
[1/3   5   6]
[  0  13  15]

I'm willing to consider changing the design to *try* to make the basis matrix over ZZ if posible, otherwise make it over QQ. Thoughts? I doubt this will break anything.... (famous last words).

[jason]

As per request on IRC:

[23:17] <ncalexan> jason--: I think that decision is fine.  I say close it as invalid.