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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?

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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.

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  • $\begingroup$ Is there a proof of this fact? $\endgroup$ – Happy Oct 6 '18 at 17:11
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    $\begingroup$ 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. $\endgroup$ – Ted Shifrin Oct 6 '18 at 17:16
  • $\begingroup$ What about the rest of my questions? $\endgroup$ – Happy Oct 6 '18 at 17:40
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    $\begingroup$ I already answered them. :) $\endgroup$ – Ted Shifrin Oct 6 '18 at 17:41
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    $\begingroup$ As I said, any vector space is a manifold because you have a global parametrization. You need to understand the definitions! $\endgroup$ – Ted Shifrin Oct 6 '18 at 17:54

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