e = \{(1,0,0), (0,1,0), (0,0,1)\}, [A]_e^e = \left[ \begin{array}{rrr} 1 & 1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{array} \right]\\
 f = \{(1,1,0), (1,-1,1), (0,0,-1)\}, [A]_f^f = [I_{\mathbb{R}^3}]_e^f [A]_e^e [I_{\mathbb{R}^3}]_f^e\\
 $[I_{\mathbb{R}^3}]_f^e = \left[ \begin{array}{rrr} 1 & 1 & 0 \\ 1 & -1 & 0 \\ 0 & 1 & -1 \end{array} \right]$
 $\left[ \begin{array}{rrr|rrr} 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & -1 & 0 & 0 & 1 & 0 \\ 0 & 1 & -1 & 0 & 0 & 1 \end{array} \right] \sim \left[ \begin{array}{rrr|rrr} 1 & 1 & 0 & 1 & 0 & 0 \\ 0 & -2 & 0 & -1 & 1 & 0 \\ 0 & 1 & -1 & 0 & 0 & 1 \end{array} \right] \sim \left[ \begin{array}{rrr|rrr} 2 & 2 & 0 & 2 & 0 & 0 \\ 0 & -2 & 0 & -1 & 1 & 0 \\ 0 & 2 & -2 & 0 & 0 & 2 \end{array} \right] \sim \left[ \begin{array}{rrr|rrr} 2 & 0 & 0 & 1 & 1 & 0 \\ 0 & -2 & 0 & -1 & 1 & 0 \\ 0 & 0 & -2 & -1 & 1 & 2 \end{array} \right]$\\
 $\displaystyle \Rightarrow \ [I_{\mathbb{R}^3}]_e^f = \frac{1}{2} \left[ \begin{array}{rrr} 1 & 1 & 0 \\ 1 & -1 & 0 \\ 1 & -1 & -2 \end{array} \right]$
 \begin{align*}
 [A]_f^f & = \frac{1}{2} \left[ \begin{array}{rrr} 1 & 1 & 0 \\ 1 & -1 & 0 \\ 1 & -1 & -2 \end{array} \right] \left[ \begin{array}{rrr} 1 & 1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{array} \right] \left[ \begin{array}{rrr} 1 & 1 & 0 \\ 1 & -1 & 0 \\ 0 & 1 & -1 \end{array} \right] \\
 & = \frac{1}{2} \left[ \begin{array}{rrr} 1 & 2 & -1 \\ 1 & 0 & 1 \\ 1 & 0 & -1 \end{array} \right] \left[ \begin{array}{rrr} 1 & 1 & 0 \\ 1 & -1 & 0 \\ 0 & 1 & -1 \end{array} \right] = \frac{1}{2} \left[ \begin{array}{rrr} 3 & -2 & 1 \\ 1 & 2 & -1 \\ 1 & 0 & 1 \end{array} \right]
 \end{align*}