Can I claim my 36 y/o girlfriend as a dependent since I am head of house and she does not work?

No. 36 years old and not married? How old are you? I don't recommend living with someone unless you are married. You can claim zero on your taxes and get more back at tax time.

IRS definition of dependent

Who Can Claim a Dependent's Expenses in Pub. 970, chapter 2 or 3.

http://www.irs.gov/pub/irs-pdf/p970.pdf

If you provide over 50% of her total support (housing, food, medical, school, clothing, etc.), her income is less than $3,500, and she has lived with you the entire year, yes.

Edit: The incorrect answers here are the reason you shouldn't depend on strangers to answer your questions.

Oh, the source:

http://www.irs.gov/publications/p501/ar02.html

Absolutely not. She is considered a room mate & you can't claim a room mate. This is one of the reasons people want to get married.

Yes

@V2, that would be funny adopt a 36 yr old. ROFL

Of course. You also can adopt her.