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]
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