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In linear algebra, an orthogonal transformation is a linear transformation T : V → V on a real inner product space V, that preserves the inner product. That is, for each pair u, v of elements of V, we have
Since the lengths of vectors and the angles between them are defined through the inner product, orthogonal transformations preserve lengths of vectors and angles between them. In particular, orthogonal transformations map orthonormal bases to orthonormal bases.
Orthogonal transformations are injective: if T v = 0 {\displaystyle Tv=0} then 0 = ⟨ T v , T v ⟩ = ⟨ v , v ⟩ {\displaystyle 0=\langle Tv,Tv\rangle =\langle v,v\rangle } , hence v = 0 {\displaystyle v=0} , so the kernel of T {\displaystyle T} is trivial.
Orthogonal transformations in two- or three-dimensional Euclidean space are stiff rotations, reflections, or combinations of a rotation and a reflection. Reflections are transformations that reverse the direction front to back, orthogonal to the mirror plane, like mirrors do. The matrices corresponding to proper rotations have a determinant of +1. Transformations with reflection are represented by matrices with a determinant of −1. This allows the concept of rotation and reflection to be generalized to higher dimensions.