# Why $T_{A}M(n) = M(n)$ and why $T_{f(A)}S(n) = S(n)$ and why $M(n)$ & $S(n)$ are manifolds. (Guillemin & Pollack p.23)

Why $$T_{A}M(n) = M(n)$$ and why $$T_{f(A)}S(n) = S(n)$$ and why $$M(n)$$ & $$S(n)$$ are manifolds?

$$M(n)$$ is the space of all $$n x n$$ matrices and $$S(n)$$ is the space of all $$n x n$$ symmetric matrices.

A manifold definition is: a subset $$X$$ of some ambient euclidean space $$R^{N}$$ is a k-dimensional manifold if it is locally diffeomorphic to $$R^k$$.

Could anyone clarify the above questions for me please?

If you have a vector space $$V$$, for any element $$v\in V$$, we always have $$T_vV = V$$. The space $$M(n)$$ of matrices is a vector space, and $$S(n)$$ is a vector subspace.
• Sure: You have a global parametrization $\phi\colon \Bbb R^n\to V$ when $V$ is $n$-dimensional, and $T_vV = \text{im}(d\phi_a)$ when $\phi(a)=v$. But $\phi$ is a vector space isomorphism. You finish. – Ted Shifrin Oct 6 '18 at 17:16