Do professors change their impression of a previously bad student? One of my friends (a math graduate student from my department) didn't work hard during his undergraduate studies, and his GPA was, naturally, low. He barely (and with some luck) got into grad school, where he works harder.

The problem is that as it seems, most of the professors have a very bad impression of him, both because some of them had him in their classes, and because our department is relatively small and professors exchange a lot of information and opinions about students.

Part of the problem is that his reputation is not (just) of someone who is not hard-working (that would have been easier to fix, perhaps), but at least some of the professors simply don't consider him as smart or talented enough, and that is based solely on courses he participated in -- neither of those professors had any actual mathematical interaction or deep conversation with him, except for his advisor (who knows he wasn't serious enough about his studies) and perhaps a few others, who seem to have a somewhat better opinion of him than most.

Is there a way for this student to change the impression people have of him? From your experience, do people really change their impression of someone, once new information is presented (or are they likely to still think of him as not smart enough to be a mathematician, but who manages to go through grad school with hard work)?

This question might be related, How to change the idea that supervsiors got about you? however I find the situation different: First of all, an advisor has a lot of interaction with the student, so he is likely to notice a change. Also, in the case of my friend, the advisor does believe in his potential, but probably doesn't share his thoughts with people since he's not one of those who talk about students with others, and mostly keeps to himself.
Your friend needs to involve himself in department life. If he is just as invisible as his supervisor, no other professor is going to change their opinion unless they somehow get a new impression (or forget who he was).

One of the best things your friend could do is to give a department seminar if he has some interesting results to present. Make sure that it is well written and rehearsed! He should also chat with professors during social occasions, turn up at department talks and ask interesting questions; things like that.