The [[Matrix Lie group]], $\mbox{SO}(n)$ is a subgroup of [[O(n)]] and the special group of orthogonal matrices with [[Determinants]] 1. Matrices in these groups act as rotations and reflections of vectors in $\mathbb{R}$. The [[Lie algebras]] of $\mbox{SO}(n)$ is _equivalent_ to the Lie Algebra of [[O(n)]] since the [trace](Trace.md) of an [orthogonal matrix](Orthogonal%20operators.md#Orthogonal%20matrices) is $0$ regardless of the determinant of its exponentiation. #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups/Algebras/LieAlgebras #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups