Kernel/Image expression 
Express the kernel of the 1 × 4 matrix A = \begin{bmatrix}1&2&3&4 \end{bmatrix}
  as the
  image of a 4 × 3 matrix B.

I understand that the kernel of a matrix is solving the system for A$\vec{x}$ = 0, but I have no idea what this question is asking nor how to do it. Are there any kind souls who can walk me through it?

$Ax=0\Rightarrow [1 \ 2 \  3 \  4][x \  y \  z \  v ]^t=0\Rightarrow x+2y+3z+4v=0\Rightarrow x=-2y-3z-4v$
$\Rightarrow \begin{pmatrix} 
x  \\
y \\
z \\
v 
\end{pmatrix}=\begin{pmatrix} 
-2y  \\
y \\
0 \\
0 
\end{pmatrix}+\begin{pmatrix} 
-3z  \\
0 \\
z \\
0 
\end{pmatrix}+\begin{pmatrix} 
-4v  \\
0 \\
0 \\
v 
\end{pmatrix}\Rightarrow ker(A)=<\begin{pmatrix} 
-2  \\
1 \\
0 \\
0 
\end{pmatrix},\begin{pmatrix} 
-3  \\
0 \\
1 \\
0 
\end{pmatrix},\begin{pmatrix} 
-4  \\
0 \\
0 \\
1 
\end{pmatrix}>
$
Can you continue now?
