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By Mark Cartwright (auth.)

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Clearly, the result of each sequence is the same. It follows that the compositions are equal permutations. 6 30 y, (c)~ o y and (d) a o (~ o y). o ~' PERMUTATIONS Second, there is an identity for composition. The identity permutation, sometimes denoted by the Greek letter L 'iota', is the permutation that puts everything where it started. 7 Convince yourself that these operations L really do act as identities. ) The third notable property of composition is that there is an 'opposite' operation; given any permutation, there is a permutation that 'undoes' its effects.

0 -1 Finally, the identity transformation - the operation of leaving every point where it is- corresponds to the identity matrix: (1, 0) goes to (1, 0), (0, 1) to (0, 1), so the matrix is [ ~ ~ ]. It might be hoped that every isometry can be represented by a matrix in this way. Alas, this is not so. Translations cannot; nor can rotations around points other than the origin, or reflections through axes not passing through the origin. d) = (0, 0) 45 GROUPS In practice, this is not a serious problem.

A reflection in they-axis, for instance, sends (1, 0) to (-1, 0). The point (0, 1) is on the y-axis, so this reflection leaves it at (0, 1). The matrix representing this reflection is therefore: [-~ ~] For a general point (x, y), we can calculate that (x, y) [-10 0]1 = (-x, y), which is exactly where this reflection sends (x, y). 6 What matrix corresponds to the operation of reflection through the x-axis? (Hard) Show that the matrix corresponding to the operation of re. flectiOn through the hne .

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