How to find a system of equations when the solution is given?

[Lotto]

TL;DR Summary

I have to find a system of equations with this solution ${(1,2,0,3)^T+t(1,1,1,-2)^T+s(1,-1,3,0)^T;s,t \in \mathbb{R}}$ when we know that matrix of this equation has:

1. two non-zero rows
2. 3 non-zero rows.

My idea is that I could somehow use the fact that $t(1,1,1,-2)^T+s(1,-1,3,0)^T$ is the homogenous solution of the system. But what to do? Any hints?

 

[fresh_42]

You have a plane $P$ in 4D space $\mathbb{R}^4.$ The homogenous solution $P_0$ is a parallel plane where the origin has been moved from $\vec{v}_0=(1,2,0,3)^T$ to $(0,0,0,0)^T.$ The other two vectors $\vec{v}_1\, , \,\vec{v}_2$ are the directions that span the plane.
$$
\vec{x}=\vec{v}_0 + P= \vec{v}_0 + \operatorname{span}\{\vec{v}_1,\vec{v}_2\}
$$
Your equation is the parameterized description of the plane. You could for instance choose combinations like $(s,t)\in \{(0,1),(1,0)\}$ to get three points on the plane which also characterizes $P$.

I'm not quite sure whether the problem statement asks you to describe $P$ as a linear transformation of the standard basis $\{\vec{e}_1,\vec{e}_2,\vec{e}_3,\vec{e}_4\}$ in $\mathbb{R}^4$ into a basis $\{\vec{v}_1,\vec{v}_2,\vec{e}_3,\vec{e}_4\}$ (or the other way around. I tend to confuse the directions of basis transformations. I struggle between whether the bases are transformed or the subspaces), ...

... or to describe $P_0$ as the solution of $A\cdot \vec{x}=\vec{0}.$