Identify $ \mathbb{R}^4$ with the space of $2×2$ matrices $M(2×2,\mathbb{R})$.
The set $M$ of matrices with determinant $3$ is a smooth manifold of dimension $3$.
Prove that the tangent space to M at I ( identity matrix) may be identified with the set of matrices with zero trace .
How can i show this , i think i need to show every matrices with zero trace can be seen as tangent vector but how ? In addition i can't see necessity of matrices with zero trace why only matrices with zero trace.
Hints will be better to start for me .
What Sandeep is saying is that the statement, "The tangent space to M is the trace 0 matrices" is not correct. Rather than M, you need to look at a subspace of M, the set of 2 x 2 matrices with determinant 1. If you do that, parameterize a curve, and see what you can say about the determinant of a tangent vector to that curve.