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.
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