If you are just studying at the library then you could invite her to go grab a sandwich at the Cougar Eat or to play racquetball or to a movie at the international cinema or to a basketball game or bowling as part of the hanging out.

It wouldn't be as formal as a traditional date which may help you ease into more traditional dating.

I'm just an old man but to me there is a very fuzzy line between dating and hanging out. When I "dated" my wife I can only recall asking her out a few times but we started hanging out a ton and talking a ton and the "dates" sort of became implied each weekend and now "she's stuck with me" (her words, not mine).