: Mathematics: orthogonal transformation

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orthogonal transformation

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
${\\displaystyle \\langle u,v\ angle =\\langle Tu,Tv\ angle \\,.}$
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 ${\\displaystyle Tv=0}$ then ${\\displaystyle 0=\\langle Tv,Tv\ angle =\\langle v,v\ angle }$ , hence ${\\displaystyle v=0}$ , so the kernel of ${\\displaystyle T}$ is trivial.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Orthogonal_transformation)