Answer
See below
Work Step by Step
Consider set $\{(2,-1,0,1),(1,0,-1,2),(0,3,1,2),(-1,1,2,1)\} \in R^4$
Obtain $\begin{bmatrix}
2 & 1 & 0 & -1 \\-1 & 0 & 3 & 1\\0 & -1 & 1 & 2\\1 & 2 & 2 & 1
\end{bmatrix} \approx \begin{bmatrix}
2 & 1 & 0 & -1 \\-1 & 0 & 3 & 1\\0 & -1 & 1 & 2\\0 & 2 & 5 & 2
\end{bmatrix} \approx \begin{bmatrix}
0 & 1 & 6 & 1 \\-1 & 0 & 3 & 1\\0 & -1 & 1 & 2\\ 0 & 2 & 5 & 2
\end{bmatrix}$
We have the system: $\det (A)=0\begin{vmatrix}
0 & 3 & 1\\
-1 & 1 & 2\\2 & 5 & 2
\end{vmatrix}-(-1)\begin{vmatrix}
1 & 6 & 1\\
-1 & 1 & 2\\2 & 5 & 2
\end{vmatrix}+0\begin{vmatrix}
1 & 6 & 1\\
0 & 3 & 1\\2 & 5 & 2
\end{vmatrix}-0\begin{vmatrix}
1 & 6 & 1\\0 & 3 & 1\\
-1 & 1 & 2
\end{vmatrix}=(2-10)-6.(-2-4)+(-5-2)=21$
Since $\det (A)\ne 0$,the set $\{(2,-1,0,1),(1,0,-1,2),(0,3,1,2),(-1,1,2,1) \}$ is linearly independent in $R^4$
