Let $V$ be a 3d vector space with a chosen basis $\alpha=\{e_1,e_2,e_3\}, \ \beta=\{f_1,f_2,f_3\}$ for $V$ satisfying:
$$\begin{align}e_1 &= f_1+f_2+f_3 \\ e_2 &= f_2+2f_3 \\ e_3 & = f_3 \end{align}.$$
Find the four basis-change matrices:
1) $P_{\alpha}^{\beta}$ which writes $\alpha$ in terms of $\beta$
2) $P_{\beta}^{\alpha}$ which writes $\beta$ in terms of $\alpha$
3) $Q_{\alpha}^{\beta}$ which satisfies $(v)_\beta =Q_{\alpha}^{\beta}(v)_\alpha$
4) $Q_{\beta}^{\alpha}$ which satisfies $(v)_{\alpha} =Q_{\beta}^{\alpha}(v)_{\beta}$
I have listed the matrix regarding to the given information:
$$\begin{bmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 2 & 1 \end{bmatrix}$$
I do not know what is the difference between first two questions and last two. Can someone please list all the solving steps or at least give me some hint?
Thank you so much!