Dualization of matrix inequalities

Question

I am studying Linear Matrix Inequalities (LMI) in Control Theory with this lecture notes: https://www.imng.uni-stuttgart.de/mst/files/LectureNotes.pdf

I am at the Dualization Section (4.4.1, page 106) I have not yet understood these points and I hope you can help me with this. Thank you in advance!

1. First point:

Could you explain to me how the author can obtain the observation in the red box?

2. Second point: In the proof at the end of the following page, it reads

Maybe it is obvious to you but I do not get the step shown in the box. What is the property of Kernel and Image of a linear matrix used in this case? If it is ker A = im A* then it will go back to the step before that.

Thank you for your time.