Answer
u is not in the plane spanned by the columns of A.
Work Step by Step
We check whether $\mathrm{A}\mathrm{x}=\mathrm{u}$ has a solution.
If it does, then $\mathrm{u}$ is a linear combination of the columns of A.
If so, then u belongs to the plane spanned by the columns of A.
Reduce the augmented matrix $[\mathrm{A}\ \mathrm{x}]$
$\left[\begin{array}{llll}
5 & 8 & 7 & 2\\
0 & 1 & -1 & -3\\
1 & 3 & 0 & 2
\end{array}\right]\left(\begin{array}{l}
\mathrm{r}_{1}\leftrightarrow \mathrm{r}_{3}.\\
.\\
.
\end{array}\right)\sim\left[\begin{array}{llll}
1 & 3 & 0 & 2\\
0 & 1 & -1 & -3\\
5 & 8 & 7 & 2
\end{array}\right]\left(\begin{array}{l}
.\\
.\\
-5\mathrm{r}_{1}.
\end{array}\right)\sim$
$\sim\left[\begin{array}{llll}
1 & 3 & 0 & 2\\
0 & 1 & -1 & -3\\
0 & -7 & 7 & -8
\end{array}\right]\left(\begin{array}{l}
.\\
.\\
+7\mathrm{r}_{2}.
\end{array}\right)\sim\left[\begin{array}{llll}
1 & 3 & 0 & 2\\
0 & 1 & -1 & -3\\
0 & 0 & 0 & -29
\end{array}\right]$
The last row represents the equation 0=-29,
so the system is inconsistent.
u is not in the plane spanned by the columns of A.
