# Isometries There are two interpretations of the groups $O(n)$, $U(n)$, and $O(n-1,1)$, i.e. the [[Isometry Group|isometry groups]] of real [[Inner Product Space|inner product spaces]], spaces with [[Non-Degeneracy (Quadratic Forms)|non-degenerate]] [[Hermitian form|Hermitian forms]] and real spaces equipped with a [[Minkowski metric]]. These interpretations are: 1. They are matrices which implement a [[basis transformation]] from one [[orthogonality|orthonormal]] [[basis]] to another. 2. They are the matrix [[Group Representation|representation]] of operators which preserve the non-degenerate Hermitian form. View (1) corresponds to the passive view of transformations while view (2) corresponds to the active view of transformations.