On Wed, Jan 28, 2009 at 3:07 PM, Rob Beezer <[email protected]> wrote:
>
> A question about computing bases for kernels or null spaces of
> matrices.
>
> When working with students studying linear algebra for the first time,
> I like to construct basis vectors for the (right) kernel by working
> through the indices of the non-pivot columns of the reduced row
> echelon form of the matrix, setting these entries to zero, with the
> exception of one entry that is set to 1. Then the other entries are
> just negatives of certain entries of the echelon form matrix.
>
> This very approach is used (on the transpose) to compute the (left)
> kernel. But then it gets used in the constructor of a subspace, which
> calls a submodule constructor, which "echelonizes" the basis
> (presumably to achieve some canonical representation). This means the
> kernel produces a vector space with a very nice basis, but my students
> won't recognize it. I'd like to get the pivot/non-pivot basis back.
>
> Could the construction of the pivot/non-pivot basis be isolated from
> the kernel() code in matrix2.pyx? And maybe called nullspace_basis
> () or something similar? Then kernel() could call this method on the
> transpose, and use the result to construct the subspace.
That's a great idea.
I just looked at the kernel code in matrix2.pyx. Currently there is
only left_kernel and right_kernel. I notice with bemusement that
*both* functions transpose their input then do something! Pretty
stupid.
I think the right change for your application would be to change this
line in left_kernel:
W = V.submodule(basis)
to
if echelonize:
W = V.submodule(basis)
else:
W = V.submodule_with_basis(basis)
then put an echelonize=True option in the left_kernel function header.
William
-- William
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