Let $P = \begin{bmatrix} 1 & 2 & -1 \ -3 & -5 & 0 \ 4 & 6 & 1 \end{bmatrix}$, $v_1 = \begin{bmatrix} -2 \ 2 \ 3 \end{bmatrix}$, $v_2 = \begin{bmatrix} -8 \ 5 \ 2 \end{bmatrix}$, $v_3 = \begin{bmatrix} -7 \ 2 \ 6 \end{bmatrix}$ (1)
(a) Find a basis $U = \{u_1, u_2, u_3\}$ for $\mathbb{R}^3$ such that $P$ is the change-of-coordinates matrix from
$U$ to the basis $V = \{v_1, v_2, v_3\}$. Show all support work.
(b) Find the $U$-coordinates of $v_2$. Show all support work.
(c) Find the $V$-coordinates of $v_2$. Justify your answer.
(d) Find a basis $W = \{\omega_1, \omega_2, \omega_3\}$ for $\mathbb{R}^3$ such that $P$ is the change-of-coordinates matrix from
$V$ to $W$
