How to find matrix from general solution? 
Find a $2 \times 3$ system (two equations with three unknowns) such that its general solution has form $\begin{pmatrix}1\\1\\0 \end{pmatrix}+s\begin {pmatrix}1\\2\\1\end{pmatrix},\ s \in \Bbb R.$ 

I tried thinking that $s\begin {pmatrix}1\\2\\1\end{pmatrix}$ is solution to kernel of asked matrix, and tried matrix $\begin {pmatrix}1&-1&1\\1&-1&1\end{pmatrix}$, but it also includes $\begin{pmatrix}1\\1\\0 \end{pmatrix}$ in its kernel!

One can translate the first two components of $u= (x,y,z)^T= (s+1, 2s+1, s)^T$ into the following system $Au=b$, the third component stays free and acts as $z=s$:
$$
x = s + 1 = z + 1 \iff x - z  = 1 \\
y = 2s + 1 = 2 z  + 1 \iff y - 2 z = 1 \\
$$
Or written as augmented matrix
$$
(A\mid b) =
\left(
\begin{array}{rrr|r}
1 & 0 & -1 & 1 \\
0 & 1 & -2 & 1 \\
\end{array}
\right)
$$
