Hi,
On Thu, Sep 05, 2019 at 06:33:34PM -0400, David Joyner wrote:
> On Thu, Sep 5, 2019 at 6:29 PM David Guichard <david.guich...@gmail.com>
> wrote:
>
> > When is a linear system consistent?
> >
> > This isn't helpful:
> >
> > a,b,c=var('a b c')
> > A=matrix([[-4,5,9,a],[1, -2, 1, b],[-2,4,-2,c]])
> > A.echelon_form()
> > A.rref()
> >
> > [ 1 0 -23/3 0]
> > [ 0 1 -13/3 0]
> > [ 0 0 0 1]
> >
> > [ 1 0 -23/3 0]
> > [ 0 1 -13/3 0]
> > [ 0 0 0 1]
> >
> >
> > But this works, so I know how to get what I want:
> >
> > R.<a,b,c>=QQ[]
> > A=matrix(R,[[-4,5,9,a],[1, -2, 1, b],[-2,4,-2,c]])
> > A.echelon_form()
> >
> > [ 1 0 -23/3 -2/3*a - 5/3*b]
> > [ 0 1 -13/3 -1/3*a - 4/3*b]
> > [ 0 0 0 2*b + c]
> >
> > So [a,b,c] is in the span of the columns iff 2b+c==0.
> >
> > Somewhat oddly, rref doesn't work here:
> >
> > A.rref()
> >
> > [ 1 0 -23/3 0]
> > [ 0 1 -13/3 0]
> > [ 0 0 0 1]
> >
> >
> This is because the CAS used assumes 2*b + c is non-zero.
> I would call it an "unfortunate feature", except that it (this "feature")
> gives teachers of linear algebra examples for tests and quizzes that
> can't be correctly solved using a CAS, and have to be solved by hand.
I would say that there is a bug in the 'default' algorithm. Note that
changing the algorithm leads to:
sage: A.echelon_form(algorithm='partial_pivoting')
[ 1 0 -23/3 -2/3*a + 5/6*c]
[ 0 1 -13/3 -1/3*a + 2/3*c]
[ 0 0 0 b + 1/2*c]
Ciao,
Thierry
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