$
\newcommand{\colvec}[1]{\begin{bmatrix} #1 \end{bmatrix}}
\newcommand{\rowvec}[1]{\begin{bmatrix} #1 \end{bmatrix}}
\newcommand{\mat}[1]{{\boldsymbol{{#1}}}} % matrix
\renewcommand{\vec}[1]{{\boldsymbol{{#1}}}} % vector
$
Using Gaussian elimination, find all solutions of the inhomogeneous equation system $\mat A \vec x = \vec b$, with
\begin{align*}
\mat A =
\begin{bmatrix}
0 & 1 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 & 0 & 1
\end{bmatrix}\,,\quad
\vec b:=
\begin{bmatrix}
2\\-1\\1
\end{bmatrix}\,.
\end{align*}
\begin{align*}
\mathcal S = \left\{
\vec x\in\R^6: \vec x =
\lambda_1 \colvec{1\\0\\0\\0\\0\\0}\ +
\lambda_2 \colvec{0\\0\\1\\0\\0\\0} +
\lambda_3 \colvec{0\\-1\\0\\-1\\1\\1} +
\colvec{0\\1\\0\\-2\\1\\0}
\,,\quad\lambda_1 , \lambda_2, \lambda_3 \in\R
\right\}\,.
\end{align*}