Introduction - math.uconn.edu

ISOMETRIES OF RnKEITH a functionh:Rn Rnthat preserves the distance between vectors:||h(v) h(w)||=||v w||for allvandwinRn, where||(x1,..,xn)||= x21+ + identity transformation: id(v) =vfor allv : id(v) = vfor allv : fixingu Rn, lettu(v) =v+u. Easily||tu(v) tu(w)||=||v w||.Example around points and reflections across lines in the plane are isome-tries ofR2. Formulas for these isometries will be given in Example and Section effects of a translation, rotation (around the origin) and reflection across a line inR2are pictured below on sample line composition of two isometries ofRnis an isometry. Is every isometry invertible? Itis clear that the three kinds of isometries pictured above (translations, rotations, reflections)are each invertible (translate by the negative vector, rotate by the opposite angle, reflect asecond time across the same line).In Section 2, we show the close link between isometries and the dot product onRn,which is more convenient to use than distances due to its algebraic properties.

2 KEITH CONRAD 2. Isometries and dot products Using translations, we can reduce the study of isometries of Rnto the case of isometries xing 0. Theorem 2.1. Every isometry of Rncan be uniquely written as the composition t kwhere tis a translation and kis an isometry xing the origin.

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