Why would my ex girlfriend be jealous?

So more than 3 months ago, my ex-girlfriend broke it off with me saying she needed space. I agreed and didn't bother to fight for her back. Whatever. Two weeks after she called it off, she started to hang around with someone who she didn't like. (she told me she didn't like him when she was with me). I know she's trying to get a reaction out of me so whatever. I'm ignoring it.

I see her a lot as we do go to the same gym. The other night, I was having a conversation with a female friend (She knows of me and my ex). My ex and her friend couldn't stop staring at us. According to my friend, my ex's back would be facing us but her eyes would be locked on us. Observing us. Then the following day, my ex seems to try and grab my attention again playing these games.

She's always hot and cold during our break up. One day friendly the next day cold. Why?

What is going on here? My friend said our vibe with each other is very platonic.
Why would my ex girlfriend be jealous?
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