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This is a homework to find the connected components of $GL(n),O(n),U(n)$.

$GL(n),O(n)$
There is a hint about this. $GL(n),O(n)$ has two connected components $GL_{+}(n),GL_{-}(n)$ and $SO(n),O(n)-SO(n)$.
I think it just divide the $det$ of matrix into two parts: one is positive and the other is negative. So we can prove the positive one is not connected with the negative one because if we make it, there would be a continuous function $\sigma:I\rightarrow{GL(n)}$ and $det(\sigma(0))<0,det(\sigma(1))>0$. So I get stuck here. How to next?

$U(n)$
How is it going on $U(n)$? It is not similar to the situation above.

I hope someone can help me. Thank you!

gaoxinge
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    1: $(-\infty,0)$ and $(0,\infty)$ are not connected in $\mathbb{R}.$ Think about what continuous map does with connected components. 2: The fact $SO(n)$ is (path) connected can be seen by interpreting as a space of rotations. 3: Note that $GL_+(n)$ can be deformed to $SO(n)$. – Braindead Jan 12 '14 at 07:27

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You are right about $\det$. For $SL(n)$ or $GL_+(n)$ see this thread Prove that the set of $n$-by-$n$ real matrices with positive determinant is connected

For $U(n)$ write your matrix in the form $ADA^{-1}$, where $A$ is unitary and $D$ is unitary diagonal. Show, that you can connect $D$ to $I$ in diagonal matrices (that's easy)

For SO(n) see this thread proving that $SO(n)$ is path connected

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