To find an explicit description of the null space, also referred to as [tex]\( \text{Nul } A \)[/tex], we need to determine the vectors that span this space. Let's start with the given matrix [tex]\( A \)[/tex]:
[tex]\[
A = \begin{pmatrix}
1 & 4 & 3 & 0 \\
0 & 1 & 2 & -8
\end{pmatrix}
\][/tex]
The null space of a matrix [tex]\( A \)[/tex] consists of all the vectors [tex]\( x \)[/tex] such that [tex]\( A x = 0 \)[/tex].
The vectors that span the null space of [tex]\( A \)[/tex] are given as:
[tex]\[
\begin{pmatrix}
5 \\
-2 \\
1 \\
0
\end{pmatrix},
\begin{pmatrix}
-32 \\
8 \\
0 \\
1
\end{pmatrix}
\][/tex]
Thus, a spanning set for [tex]\( \text{Nul } A \)[/tex] is:
[tex]\[
\left\{
\begin{pmatrix}
5 \\
-2 \\
1 \\
0
\end{pmatrix},
\begin{pmatrix}
-32 \\
8 \\
0 \\
1
\end{pmatrix}
\right\}
\][/tex]
So the explicit description of [tex]\( \text{Nul } A \)[/tex] is written with the spanning vectors as:
[tex]\[
\left\{ \begin{pmatrix} 5 \\ -2 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} -32 \\ 8 \\ 0 \\ 1 \end{pmatrix} \right\}
\][/tex]
Therefore, a spanning set for [tex]\( \text{Nul } A \)[/tex] is [tex]\(\boxed{\left\{ \begin{pmatrix} 5 \\ -2 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} -32 \\ 8 \\ 0 \\ 1 \end{pmatrix} \right\}}\)[/tex].
