Should I have asked her out?

Hello, I'm having a situation I would like some advice on. I am an older graduate student at a college, and I work as a teaching assistant. Before anyone gets any ideas, no it wasn't one of my students that I asked out, however it was a younger girl who is also on my level as a graduate student at this same college. When I say older, I am in my 30's whereas other graduate students tend to be in their early to mid 20's. In this case I believe I'm about 7-8 years older than the girl I asked out (I don't know for sure). Basically what happened was that this girl, let's call her G, left an anonymous cutesy note in my mailbox. I don't want to go into detail what the note said lest I somehow be identified through it in the future, but in short it expressed something along the lines of how great she thought I was and had a cute drawing with a smilie face included as well. I had for sure thought it was one of my students that had left it there after what I figured was just a satisfying lecture that I had given. So I made a joke in class about how I was thanking the anonymous student that left it there, but nobody really owned up to giving me that letter. Within a few days G came up to me and admitted that she had written the letter and left it in my mailbox. Well to be honest, I was rather confused as I was very convinced it had simply been one of my students. I didn't know why she had written it, and when I asked her she just said it was a spur of the moment thing because she happened to be in the mailroom and saw my mailbox. Either way, I felt this left me in an awkward situation. I felt that if I ignored the fact that she had written it, and that she had admitted to writing it, then I was ignoring her very obvious hint that she liked me and not asking her out would've basically been a rejection to her. But at the same time I knew I was very much older than her, and she could've very well just done this as a joke without a second thought. It should be said that the two of us are barely even friends. We don't hang out, we don't chat outside the University. We only had a few classes together at the start of our grad program and right now we're both working on our thesis so we only really see each other if we randomly run into one another in the grad lounge. Basically I don't know much about her, and as far as I know she doesn't know much about me. But I figured, I wasn't against the idea of a relationship with her if she was hinting she wanted to give it a shot. I thought about it for a few days, and next time I ran into her (and thankfully it was just the two of us at the time) I flat out asked her out, and she said no. Well, okay, problem solved. I didn't want her to think I wasn't attracted to her after she gave me the letter, but at the same time I was wary of a relationship due to the age difference. She had said no, and to me that was that. She really had given me the letter as a spur of the moment decision, and I could move on. Fast forward about two months, and a mutual friend that we know (more her friend than mine), points out to me that my asking G out made G very uncomfortable. G felt that I had hit on her inappropriately and that I was very weird and creepy to be asking her out considering how much older I am than she is. G also apparently said that she would be making an effort to avoid me because of this situation. Thinking back, she may have actually been avoiding me since we only ran into each other once in those two months, and we hadn't exchanged much more than a greeting during that brief encounter. Knowing this is in turn making ME feel very uncomfortable. I'm not sure how to move forward and deal with this situation. I feel as if I should apologize for asking her out, but I'm not sure approaching her again after knowing what I know is a smart decision. I just don't understand how it came to this. Was my reasoning in asking her out wrong? Should I not have assumed she was hinting at something with that letter and just taken her words at face value? I'm hoping for any advice out there that someone may offer, and maybe even some reassurance that I didn't act unreasonably. I feel like I could really use that right about now. This whole situation has just deeply disturbed me and I keep blaming myself for not just keeping my mouth shut and letting that stupid letter slide.

Hey! First of all i dont think youre in the wrong at all! she initiated with the letter and if it were me id assume someone liked me if they wrote a letter to me. asking her out wasnt a bad move either at least you knew how serious she was. I think shes just regretted her decision and felt embarrassed about it and thats why she shot you down and thats why she will have told her friends that stuff about you. I think you have 2 options just have to ignore her and not get het up about it or approach her and apologize and explain why you asked her out. at the end of the day its her loss not yours! dont feel uncomfortable about it as you did nothing wrong! :)