Find basis of solutions of this linear system

Supposed to find basis of the subspace of vector space ${\mathbb{R}}^{3}$ of solutions of this linear system of equations:

$y=\{\begin{array}{l}{x}_{1}+2{x}_{2}-{x}_{3}=0\\ 2{x}_{1}+7{x}_{2}-2{x}_{3}=0\\ -{x}_{1}+3{x}_{2}+{x}_{3}=0\end{array}$

I solve this system and I got: ${x}_{1}={x}_{3}$ and ${x}_{2}=0$

$\overrightarrow{x}=\left[\begin{array}{c}{x}_{1}\\ 0\\ {x}_{1}\end{array}\right]={x}_{1}\left[\begin{array}{c}1\\ 0\\ 1\end{array}\right]+0\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$

Is the basis: $\left[\begin{array}{c}1\\ 0\\ 1\end{array}\right]$?

Supposed to find basis of the subspace of vector space ${\mathbb{R}}^{3}$ of solutions of this linear system of equations:

$y=\{\begin{array}{l}{x}_{1}+2{x}_{2}-{x}_{3}=0\\ 2{x}_{1}+7{x}_{2}-2{x}_{3}=0\\ -{x}_{1}+3{x}_{2}+{x}_{3}=0\end{array}$

I solve this system and I got: ${x}_{1}={x}_{3}$ and ${x}_{2}=0$

$\overrightarrow{x}=\left[\begin{array}{c}{x}_{1}\\ 0\\ {x}_{1}\end{array}\right]={x}_{1}\left[\begin{array}{c}1\\ 0\\ 1\end{array}\right]+0\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$

Is the basis: $\left[\begin{array}{c}1\\ 0\\ 1\end{array}\right]$?