The linear algebra routines implemented in GEL do not currently come from a well tested numerical package, and thus should not be used for critical numerical computation. On the other hand, Genius implements very well many linear algebra operations with rational and integer coefficients. These are inherently exact and in fact will give you much better results than common double precision routines for linear algebra.
For example, it is pointless to compute the rank and nullspace of a floating point matrix since for all practical purposes, we need to consider the matrix as having some slight errors. You are likely to get a different result than you expect. The problem is that under a small perturbation every matrix is of full rank and invertible. If the matrix however is of rational numbers, then the rank and nullspace are always exact.
In general when Genius computes the basis of a certain vectorspace
(for example with the
NullSpace) it will give the basis as
a matrix, in which the columns are the vectors of the basis. That is, when
Genius talks of a linear subspace it means a matrix whose column space is
the given linear subspace.
It should be noted that Genius can remember certain properties of a
matrix. For example, it will remember that a matrix is in row reduced form.
If many calls are made to functions that internally use row reduced form of
the matrix, we can just row reduce the matrix beforehand once. Successive
rref will be very fast.