Answer
Nul A= Span $\{ \left[\begin{array}{l}
7\\
-4\\
1\\
0
\end{array}\right],\ \left[\begin{array}{l}
-6\\
2\\
0\\
1
\end{array}\right] \}$
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 $0]$ = $\left[\begin{array}{lllll}
1 & 3 & 5 & 0 & 0\\
0 & 1 & 4 & -2 & 0
\end{array}\right] \left(\begin{array}{l}
R_{1}=R_{1}-3R_{2}\\
.
\end{array}\right)$
$\sim\left[\begin{array}{lllll}
1 & 0 & -7 & 6 & 0\\
0 & 1 & 4 & -2 & 0
\end{array}\right]$
With $x_{3}$ and $x_{4}$ free parameters (any real numbers),
$x_{1}=7x_{3}-6x_{4}$
$x_{2}=-4x_{2}+2x_{4}$
$x=\left[\begin{array}{l}
7x_{3}-6x_{4}\\
-4x_{3}+2x_{4}\\
x_{3}\\
x_{4}
\end{array}\right]=x_{3}\left[\begin{array}{l}
7\\
-4\\
1\\
0
\end{array}\right]+x_{4}\left[\begin{array}{l}
-6\\
2\\
0\\
1
\end{array}\right]$
Nul A= Span $\{ \left[\begin{array}{l}
7\\
-4\\
1\\
0
\end{array}\right],\ \left[\begin{array}{l}
-6\\
2\\
0\\
1
\end{array}\right] \}$
(the answer in the back of the book is different, but this is correct)
