What are xi and v?
The symbols xi and v were used in proving that if the Reachability Gramian W is full rank then the system is reachable. This is done by choosing an arbitrary xi as the desired final state that we want to ask whether we can reach with some control input. The variable v = inv(W)*xi, which is simply an intermediate variable in the proof that is related to the control u (and thus tells us something about the direction that we need to push to move the system to state xi, but it is the control u itself that we want to know). The pdf of the lecture notes are available online.