for Question 2 d) can someone point out where my working went wrong:
Here is my working:
every vector in $\mathbb{R}^3$ after applying basis $B$ is: $\begin{align} \begin{bmatrix} x_{1}+x_3 \\ x_1+x_{2} \\ x_{1}+x_3 \end{bmatrix} \end{align}$. Obtained by applying $ \left[ \begin{array}{ccc} 1&0&1\\ 1&1&0\\ 0&1&1 \end{array}\right]$ to $\begin{align} \begin{bmatrix} x_{1}\\ x_{2} \\ x_3 \end{bmatrix} \end{align}$.
Similarly, every vector in $\mathbb{R}^2$ after applying basis $A$ is: $\begin{align} \begin{bmatrix} 2x_{1}-x_2+x_3 \\ -x_2+x_3 \end{bmatrix} \end{align}$.
So, ${}_A[f]_B = \begin{align} \begin{bmatrix} 2&0&-1\\ 2&-2&1 \end{bmatrix} \end{align}$ by $ \begin{align} \begin{bmatrix} 2&0&-1\\ 2&-2&1 \end{bmatrix} \begin{bmatrix} x_{1}+x_3 \\ x_1+x_{2} \\ x_{1}+x_3 \end{bmatrix} = \begin{bmatrix} 2x_{1}-x_2+x_3 \\ -x_2+x_3 \end{bmatrix} \end{align} $.
the correct answer is : ${}_A[f]_B = \begin{align} \begin{bmatrix} \frac{1}{2}&0&\frac{3}{2}\\ -\frac{1}{2}&1&\frac{3}{2} \end{bmatrix} \end{align}$