A is a matrix not a subspace. I think you left out the words 'column space'? You should also be able to say something stronger than 'subspace' at this point-remember the bit about the ranks.
A general thing about projections- if T is a projection onto a subspace U, and v is any vector in U then T(v)=v. You should be able to prove this. You should then be able to prove (AA')v=v where v is any column of A.