"practice 7 problem 14c why for the X1 we use an eigenvector from the first rank?"

The Jordan normal form requires the correct number of linearly independent generalised eigenvectors.

If we can get all the linearly independent eigenvectors at rank 1 then we have already finished our mission.

If not then we raise the rank to the highest order and apply the recursive formula given in the lecture to get specifically related eigenvectors.

Now I did not show you where the recursive formula comes from or why it works. As I mentioned this was intended only as a brief overview of Jordan normal form. There are full lectures given on this topic which I can provide resources too if you are interested.