Parametric Vector Form

In general, suppose the free variables for $A\text{x}=\text{0}$ are $x_k,\dots ,x_n$. Then all the solutions to $A\text{x}=\text{0}$ can be written as: $\text{x}=x_k{\text{v}}_k+x_{k+1}{\text{v}}_{k+1}+\cdots +x_n{\text{v}}_n$ For some ${\text{v}}_k,\dots {\text{v}}_n$.
This is Parametric Vector Form.

In RREF: $\left[\begin{array}{cccc}1& 0& -2& 1\\ 0& 1& 1& 1\\ 0& 0& 0& 0\end{array}\right]$

x_1 = 1+ 2x_3 \\ x_2=1-x_3 \\ x_3 = x_3 }

$$\begin{gathered} \mathbf{x}=\left[\text{array}1+2x_3 \\ 1-x_3 \\ 0+x_3\right]=\left[\text{array}1 \\ 1 \\ 0\right]+\left[\text{array}2x_3 \\ -x_3 \\ {x}_3\right]=\left[\text{array}1 \\ 1 \\ 0\right]+x_3\left[\text{array}2 \\ -1 \\ 1\right] \end{gathered}$$

\text{if, }\mathbf{x}= \left[ \begin{array} \\ 1+2x_3 \\ 1-x_3 \\0+ x_3 \end{array} \right] \text{in parametric vector form}, \mathbf{x}= \left[ \begin{array} \\1 \\ 1 \\0 \end{array} \right]+x_3\left[ \begin{array} \\2 \\ -1 \\1 \end{array} \right]