If they both mean the same, is が optional?
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As far as I understand, they mean the same thing. The one with
Edit: Apparently this is ungrammatical, although I don't understand why. Maybe I'll make it a separate topic.
気が付く has three meanings.
For senses 1 and 3, 気付く is also used. When used in these senses, I do not think that there is any difference in meaning between 気が付く and 気付く. I do not recognize any difference in formality, either, but I may be wrong.