Canonical way of turning implicit equations into a subspace and vice versa

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I am not sure how "canonical" it is, but i would do the following:

For your first question, satisfying an equatiion corresponds to gelonging to some kernel:

sage: M = matrix(QQ,[[1,2,3,]])
sage: M.right_kernel()
Vector space of degree 3 and dimension 2 over Rational Field
Basis matrix:
[   1    0 -1/3]
[   0    1 -2/3]

sage: M.right_kernel().basis()
[
(1, 0, -1/3),
(0, 1, -2/3)
]

So, the set of vectors that satifsfy ax+by+cz=0 are the t_1 * (1, 0, -1/3) + t_2 * (0, 1, -2/3) for t_1 and t_2 in QQ.

For your second question, you can do:

sage: v1 = vector(QQ,(1,2,3))
sage: v2 = vector(QQ,(4,5,6))

sage: V = span([v1,v2],QQ) ; V
Vector space of degree 3 and dimension 2 over Rational Field
Basis matrix:
[ 1  0 -1]
[ 0  1  2]

sage: vector((5,7,9),QQ) in  V
True