---- > [!definition] Definition. ([[special orthogonal group]]) > The **special orthogonal group** $\text{SO}(n, \mathbb{R})$ is the [[group]] of [[orthogonal matrix|orthogonal matrices]] that [[orientation-preserving and orientation-reversing operators|preserve orientation]] (i.e., have [[determinant|det]] $= +1$). $\operatorname{SO}(n, \mathbb{R})$ is a [[clopen set|clopen]] [[Lie subgroup]] of the [[general orthogonal group|orthogonal group]] given by $\operatorname{SO}(n, \mathbb{R}) := \operatorname{O}(n) \cap U,$ [[subspace topology|where]] $U$ is the open subset of $\mathbb{R}^{n \times n}$ given by $U:= \det ^{-1}( ( 0, \infty ) )$ using that [[determinant]] is [[continuous]] function. Being an open subset of $\operatorname{O}(n)$, $\operatorname{SO}(n ,\mathbb{R})$ also dimension $\frac{n(n+1)}{2}$ and has [[Lie algebra]] being that of all [[skew-symmetric matrix|skew-symmetric matrices]] in $\mathbb{R}^{n \times n}$. $\operatorname{SO}(n, \mathbb{R})$ is also [[closed set|closed]], since it also equals $\operatorname{O}(n) \cap C$, where $C:=\det^{-1}(\{ 1 \}),$ hence by the [[closed subgroup theorem]] an [[embedded submanifold|embedded]] [[Lie subgroup]] of $\operatorname{O}(n)$. It is the [[identity component of a Lie group|identity]] [[connected component]] of $\operatorname{O}(n)$; this I found trickier to prove than expected (but not too bad). > [!basicproperties] > $\mathfrak{so}(3) \cong(\mathbb{R}^{3}, \times)$[[cross product|.]] ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```