# cotangent bundles – How to prove a subspace of symmetric matrix space is a manifold

I am studying Differential Geometry and came across to this problem:

Prove that the set of all idempotent symmetric matrices $$n times n$$ of rank $$k< n$$ is a manifold and I need to find its tangent space. I know how to give a proof when it contains just symmetric matrices, but now it became a big problem to me.

I appreciate any tip or solution to this question.