5. Let $U = \{u_1, u_2, u_3\}$ and $W = \{w_1, w_2\}$, where
$u_1 = \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}$, $u_2 = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$, $u_3 = \begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix}$, $w_1 = \begin{pmatrix} 1 \\ -1 \end{pmatrix}$, $w_2 = \begin{pmatrix} 2 \\ -1 \end{pmatrix}$.
Suppose $L: \mathbb{R}^3 \to \mathbb{R}^2$ is defined by $L(x) = (x_1 + x_2, x_1 - x_3)^T$. (a) Find the matrix representation of
$L$ with respect to $U$ and $W$. (b) Use your results to compute $[L(x)]_W$, assuming $x = 2u_1 - u_2 + \frac{1}{2}u_3$.
Verify that your results are correct.