# Problem concerning smooth manifolds

A={ {{cos x, -sin x},{sin x, cos x}}|x $$\in$$R}, show that set A is smooth manifold in space of 2x2 real matrix. What is tangent space in unity matrix?

1. What is topology here? (Because I need topology to show that this is manifold)
2. In solution they say that mapping x$$\rightarrow$${{cos x, -sin x},{sin x, cos x}} is diffeomorphism between circle and given set and from this follows that given set is manifold.
How to prove that this is diffeomorphosm? (proving by definition is pretty hard, although it's somehow obvious that this is true if use proper topology)
Diffeomorphism is special kind of mapping between two smooth manifolds - so I really don't know to show that this mapping is diffeomorphism (because before that I must show that A is smooth manifold).
3. They also say that in coords of this one dimensional smooth manifold, tangent vector is $$\frac{\partial}{\partial x}$$?

Do you recognize that matrix as the rotation, in R2 about the origin through an angle $\theta$? In other words, every matrix in A corresponds to a point on the unit circle- the point to which (1, 0) would be rotated by this matrix. This is a "many to one" correspondence since sine and cosine are periodic with period $2\pi$. you might take, as your "coordinate neighborhoods", the intervals $(n\pi -\delta, (n+1)\pi - \delta)$, with the obvious coordinate functions from R to A.