Let P be the projective plane obtained by identifying antipode points on the unit sphere.

How to prove that the tangent space at $q \in P$ to the projective plane P is 2 dimensional?

My questions are 

1, P is not a submanifold of the Euclidean space and its tangent vectors are defined in terms of equivalence classes. How to show that there exist two linearly independent tangent vectors?

2 I hope someone can give me detailed, elementary proof without using more advanced facts--I am just starting out.
Thanks