Answer
Nul A= Span $\{ \left[\begin{array}{l}
-6\\
2\\
1\\
0\\
0
\end{array}\right],\ \left[\begin{array}{l}
8\\
-1\\
0\\
1\\
0
\end{array}\right],\ \left[\begin{array}{l}
0\\
0\\
0\\
0\\
1
\end{array}\right] \}$
Work Step by Step
By definition,
Nul A =$\{$x$: $ x$\in \mathbb{R}^{n}$ and Ax=0 $\}$.
---------
We find the general solution to Ax=0:
$[$A $0]$ = $\left[\begin{array}{llllll}
1 & 5 & -4 & -3 & 1 & 0\\
0 & 1 & -2 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0
\end{array}\right]\left(\begin{array}{l}
R_{1}=R_{1}-5R_{2}\\
.\\
.
\end{array}\right)$
$\sim\left[\begin{array}{llllll}
1 & 0 & 6 & -8 & 1 & 0\\
0 & 1 & -2 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0
\end{array}\right]$
With $x_{3},x_{4}$ and $x_{5}$ free parameters (any real numbers),
$x_{1}=-6x_{3}+8x_{4}$
$x_{2}=2x_{3}-x_{4}$
$x=\left[\begin{array}{l}
-6x_{3}+8x_{4}\\
2x_{3}-x_{4}\\
x_{3}\\
x_{4}\\
x_{5}
\end{array}\right]=x_{3}\left[\begin{array}{l}
-6\\
2\\
1\\
0\\
0
\end{array}\right]+x_{4}\left[\begin{array}{l}
8\\
-1\\
0\\
1\\
0
\end{array}\right]+x_{5}\left[\begin{array}{l}
0\\
0\\
0\\
0\\
1
\end{array}\right]$
Nul A= Span $\{ \left[\begin{array}{l}
-6\\
2\\
1\\
0\\
0
\end{array}\right],\ \left[\begin{array}{l}
8\\
-1\\
0\\
1\\
0
\end{array}\right],\ \left[\begin{array}{l}
0\\
0\\
0\\
0\\
1
\end{array}\right] \}$
