Send a letter to her or not?

ok so i work with this girl and for 2 months we really hit it off and liked each other hanging out all the time,making out all the time and texting each other..in the meantime she did have a bf whose aunt works with us as a manager. in the 3rd month she really wanted me in her college class saying how we could hang out on those days before and after class and have fun together. ok so i spend 100 bucks to get in the class and then as soon as im in she starts ignoring me! some guy in the class who i can tell she likes and he's in love with her she will only talk to. and me she left hanging everyday.. no goodbyes,no hugs or kisses...nothing.. so the last day of class she decides to ask the teacher if there are kayaks for 2. and he says yeah. so she goes kayaking with this guy the entire class and ignores me the whole last day of class...now is that shady to you guys too? all the while in this class i would just stay quiet and not send any signals that i was jealous at all. frankly i wasnt speaking to her either. next day at work i see her and she completely ignores me..no hi or anything. so i tell my friend about it and he decides behind my back he's gonna create a fake facebook and say he heard she sucks every guys u know what at our job! and he sent it! im positive she thinks its me that wrote that since i deleted her off fb some time ago because we got into it in a text where she said i have an unattractive quality. and then she said goodbye..so i wrote "as in never talking again? because i have something to say thats really mean but i didnt want to tell you". at that point she called me to ask and i said that was the wrong text to you and i said so is this the last time we talk? and she said if you want it to be..anyways that call ended and went nowhere. also her manager/aunt in law knows about us hanging out and gave her a talk about it so this could have something to do with the coldness too. but still .you cant text or talk to someone outside of work? now i see her at work and she wont look at me or talk to me..just seems really pissed off. so should i send her a note apologizing for whatever reason she's mad and say we need to talk? or is it a waste of time considering she screwed me over in that class by ignoring me, so technically i shouldnt apologize for anything