Answer
$(1,0,0),(0,1,0),(0,0,1)$
Work Step by Step
We are given: $x_1=(6,-3,2)\\
x_2=(1,1,1)\\
x_3=(1,-8,1)$ in $R^3$
We need to check whether $x_1$ and $x_2$ are linearly independent.
Obtain the matrices:
$\begin{bmatrix}
6 & -3 & 2\\
1 & 1 & 1 \\
1 & -8 &1
\end{bmatrix} \approx \begin{bmatrix}
1 & 1 & 1\\
6 & -3 & 2 \\
1 & -8 &1
\end{bmatrix} \approx \begin{bmatrix}
1 & 1 & 1\\
0 & -9 & -4 \\
0& -9 & 0
\end{bmatrix} \approx \begin{bmatrix}
1 & 1 & 1\\
0 & 0 & -4 \\
0& 1 & 0
\end{bmatrix} \approx \begin{bmatrix}
1 & 0 & 1\\
0 & 0 & 1 \\
0& 1 & 0
\end{bmatrix} \approx \begin{bmatrix}
1 & 0 & 1\\
0 & 1 & 0 \\
0& 0 & 1
\end{bmatrix} \approx \begin{bmatrix}
1 & 0 & 0\\
0 & 1 & 0 \\
0& 0 & 1
\end{bmatrix}$
Hence, $x_1, x_2$ and $x_3$ are linearly independent vectors. Consequently, $(1,0,0),(0,1,0),(0,0,1)$ are also orthogonal vectors.
Hence, a corresponding orthogonal basic for span $S$ is: $(1,0,0),(0,1,0),(0,0,1)$
