The question posed is as follows:
Find the representation of the linear operator $L:(R^3, R)→ (R^2, R): L[x] = \begin{bmatrix}x_1&-x_2\\x_2&-x_3\end{bmatrix}$ with respect to the basis $v_1 = \begin{bmatrix}1\\0\\0\end{bmatrix}, v_2 = \begin{bmatrix}1\\0\\1\end{bmatrix}, v_3 \begin{bmatrix}0\\1\\-1\end{bmatrix}$ of $(R^3, R)$ and the basis $w_1 = \begin{bmatrix}1\\1\end{bmatrix}, w_2 = \begin{bmatrix}1\\-1\end{bmatrix}$ of $(R^2, R)$.
I truly have no idea where to begin here. The professor did a terrible job explaining linear operators (which, as far as I can find, might actually be called linear transformations? That's the state of confusion I'm in), and frankly, because this professor hand-writes his assignments, I'm not even sure if the linear operator given is $L[x] = \begin{bmatrix}x_1-x_2\\x_2-x_3\end{bmatrix}$ or $L[x] = \begin{bmatrix}x_1&-x_2\\x_2&-x_3\end{bmatrix}$ (and I'm confused enough to not know which of those makes more sense). Any help at all, even to push me in the right direction, would be enormously appreciated.