This exercise comes from the Objective English course, and the correct answer is 'fully convinced of'. If you are 'convinced of' something you believe it.

The first choice is correct; the second choice doesn't make sense. The rule behind it involves the relationship between the Subject and the Verb. For example: "I convince you" (I=Subject, you=object, and the Subject performs the Verb, which is "convince"). You could also say: "You are convinced by me" (The situation is the same, but now You=subject, because the speaker wants to emphasize the recipient of the action, rather than the performer, which is "me"). It is the same principle in the sentences: "The book is boring" and "You are bored by the book"; or "The movie is amazing" and "You are amazed by the movie." The "ING" indicates that the book/movie is the source of the feeling, and the "ED" indicates that the feeling is happening to the Subject (you).

I´m convinced of/ that...