In Linear Algebra, I am often asked to find the solution set to some linear system. There are different ways to represent these solutions.
For example you could write it as
1 . a system of equations: $x_{1} + 2x_{2} + 3x_{3} = 0$
2. a spanning set (of vectors): $span
\left\{
\begin{pmatrix}
-2 \\
1 \\
0
\end{pmatrix}
,
\begin{pmatrix}
-3 \\
0 \\
1
\end{pmatrix}
\right\}
$
3. a solution set in parametric or vector form (similar to span):
$
x_{2}
\begin{bmatrix}
-2 \\
1 \\
0
\end{bmatrix}
+ x_{3}
\begin{bmatrix}
-3 \\
0 \\
1
\end{bmatrix}
$
(feel free to correct the terms in the examples above as well. I know what they look like just not what to call them.)
How would you describe the following representation of a solution set with the proper mathematical terminology?
$$ \begin{pmatrix} x_{1} \\ x_{2} \\ x_{3} \end{pmatrix} = \left\{ \left( \begin{array} {c} -(2x_{2}+3x_{3}) \\ x_{2} \\ x_{3} \end{array} \right) % \begin{array} {|c} \\ \\ \\ \end{array} % \begin{array} {c} x_{2}, x_{3} \in \mathbb{R} \end{array} \right\} $$
As Arnaud Mortier pointed out in a comment, that's still writing the solution in "parametric" form. But you're now using "set-builder notation" to represent the solution set.