Long title, but it pretty much describes it. I know that it has posted before but I can't seem to understand some points of the solution.
The problem:
Find all the linear systems whose solution space is the subspace of $\mathbb{R}^3$ spaned by the vectors $u=(5,6,6)$ and $v=(1,1,1)$
So if I'm correct the solution space should have the same basis as the subspace? Any ideas how to solve that problem?
They do not have to spanned by the same basis. simply, if you take $u - 5v = (0, 1, 1)$,
Still {$(0, 1, 1), (1, 1, 1)$} form a basis for the solution space.
However, you can consider the normal vector for the solution space. No matter how you choose a basis for the solution space, the normal vector is unique, up to scalar multiplication. It can be computed by the cross product of basis vectors of the solution space.
If you get the normal vector, then you can easily determine the linear system (plane).
Since it contains $(0, 0, 0)$, you don't even have to compute the constant term!