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 $A_{5×5}$ 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 $A$?
Best Answer
Hint: The kernel is the orthogonal complement of the row space. So $A$ needs row space equal to $W^\perp$.