Find a matrix $A$ such that $X$ generates the subspace $W$ (the solution space of the system $AX=0$.) Consider $W$ a subspace of $\mathbb{R}^{5}$ generated by:
\begin{align*}
X=\left \{(1,-1,0,5,1), (1,0,1,0,-2),(-2,0,-1,0,-1)\right \}
\end{align*}
Find a system of linear equations $AX=0$ such that W be the solution space of the system.
I understand that what I've to do is to find a matrix $A_{5x5}$ such that:
\begin{align*}
A \begin{pmatrix}
1\\ 
-1\\ 
0\\ 
5\\ 
1
\end{pmatrix}=\begin{pmatrix}
0\\ 
0\\ 
0\\ 
0\\ 
0
\end{pmatrix} \ \text{  ,  } \ A \begin{pmatrix}
1\\ 
0\\ 
1\\ 
0\\ 
-2
\end{pmatrix}=\begin{pmatrix}
0\\ 
0\\ 
0\\ 
0\\ 
0
\end{pmatrix} \ \text{ and, } \ A \begin{pmatrix}
-2\\ 
0\\ 
-1\\ 
0\\ 
-1
\end{pmatrix}=\begin{pmatrix}
0\\ 
0\\ 
0\\ 
0\\ 
0
\end{pmatrix} 
\end{align*}
So I think that the solution can be to propose a new system of linear equations but I'm not sure how can I do it. How can I find the matrix $A$?

Hint:  The kernel is the orthogonal complement of the row space.  So $A$ needs row space equal to $W^\perp$.
