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Prove that the special orthogonal group of matrices is compact and connected. Watch

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    I am trying to prove that the special orthogonal group SO_2(\mathbb{R}) is connected and compact

    SO(n)={n \times n matrices A \in O(n): det A=1}

    O(n)={n \times n matrices [/latex]A \in GL(n)}[/latex]


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    Compactness
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    To show compactness, I was thinking of using the Heine-Borel criterion that a subset V \subset \mathbb{R^{n}} is compact \iff V is closed and bounded.

    SO_2(\mathbb{R}) is closed since it is the preimage of 1 under det

    Is it bounded?


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    Connectedness

    To show SO_2(\mathbb{R}) is connected I must show that it does *not* split into two disjoint, non-empty open subsets. I was thinking of doing a proof by contradiction: suppose SO_2(\mathbb{R}) splits into two disjoint, non-empty open subsets...I think the 'open' condition fails. But I cannot think of a formal proof though

    Would really appreciate your help. Thanks
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    (Original post by number23)
    I am trying to prove that the special orthogonal group SO_2(\mathbb{R}) is connected and compact

    SO(n)={n \times n matrices A \in O(n): det A=1}

    O(n)={n \times n matrices [/latex]A \in GL(n)}[/latex]


    ----------


    Compactness
    -----------

    To show compactness, I was thinking of using the Heine-Borel criterion that a subset V \subset \mathbb{R^{n}} is compact \iff V is closed and bounded.

    SO_2(\mathbb{R}) is closed since it is the preimage of 1 under det

    Is it bounded?


    ----------
    Connectedness

    To show SO_2(\mathbb{R}) is connected I must show that it does *not* split into two disjoint, non-empty open subsets. I was thinking of doing a proof by contradiction: suppose SO_2(\mathbb{R}) splits into two disjoint, non-empty open subsets...I think the 'open' condition fails. But I cannot think of a formal proof though

    Would really appreciate your help. Thanks
    To show it's bounded, try using the fact that if M \in SO(2) then MM^T=I.

    As for connectedness, I'm not entirely sure but you might get somewhere identifying SO(2) as \mathbb{R}^4 or failing that a quick google search suggests you could go for path connectivity through the identity.
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    (Original post by number23)
    I am trying to prove that the special orthogonal group SO_2(\mathbb{R}) is connected and compact
    As joostan has pointed out, boundedness follows by looking at the condition for membership of the orthogonal group. Connectedness is usually proved by going via path connectedness. To show the latter one usually reaches for the fact that elements of special orthogonal groups can be represented as compositions of an even number of reflections. Then an argument connecting reflections to the identity finishes it off.
 
 
 
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