Answer
See answer below
Work Step by Step
We are given $S=\{(10,-6,5);(3,-3,2),(0,0,0),(6,4,-1),(7,7,-2)\}$ in $R^3$
Obtain the matrix $A=\begin{bmatrix}
10 & -6 & 5\\
3 & -3 & 2\\
0 & 0 & 0\\
6 & 4 & -1\\
7 & 7 & -2
\end{bmatrix} \approx \begin{bmatrix}
10 & -6 & 5\\
0 & -10 & 5\\
0 & 0 & 0\\
0 & -76 & 40\\
0 & -42 & 20
\end{bmatrix}\approx \begin{bmatrix}
10 & -6 & 5\\
0 & -10 & 5\\
0 & 0 & 0\\
0 & 0 & -2\\
0 & 0 & 1
\end{bmatrix} \approx \begin{bmatrix}
10 & -6 & 5\\
0 & -10 & 5\\
0 & 0 & -2\\
0 & 0 & 0\\
0 & 0 & 0
\end{bmatrix}$
Therefore, $S$ spans $R^3$.
