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How would I start off proving that the set $S$, of symmetric $n\times n$ matrices, is a manifold.

I tried using the definition directly by saying $M_n =$ the space of all $n\times n$ matrices.

For every $A\in M_n$ there exists open sets $U=V=M_n$ and a bijection $F: U\to V$ by $F(A)= A-A^T$ Therefore we have $F(U \cap S) = F(S)$ since $S$ is a subset of $M_n=\{0\} \cap M_n$ this is where I get stuck. Also, I know that the set of all symmetric $n\times n$ matrices is $\frac{n^2+n}{2}$, therefore that is the dimension of the manifold.

Definition: A set $M$ (subset of $\Bbb{R}^n$) is a $k$-dimensional manifold if for every $x\in M$ there exists open sets $U$, $V$ and a bijection $h:U\to V$ with $x\in U$ and $H(U \cap M) = V \cap (\Bbb{R}^k \times \{c^{k+1},\ldots ,c^n\})$ for all $c$'s constants

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    I don't mean to imply it's impossible to do the task you've got set-out in front of you. Only, it will be much harder to grasp the essential ideas as it will take more effort to unwind the concepts into terms you're familiar with.2010-11-03

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