College Algebra Exam Review 7

College Algebra Exam Review 7 - plication tables match up...

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1.5. PERMUTATIONS 17 The permutation that moves the object in position 1 to position 2, that in position 2 to position 3, and that in position 3 to position 1 is denoted by ± 1 2 3 2 3 1 ² : With this notation, the six permutations of three objects are ± 1 2 3 1 2 3 ² ± 1 2 3 2 3 1 ² ± 1 2 3 3 1 2 ² ± 1 2 3 2 1 3 ² ± 1 2 3 1 3 2 ² ± 1 2 3 3 2 1 ² : The product of permutations is computed by following each object as it is moved by the two permutations. If the first permutation moves an object from position i to position j and the second moves an object from position j to position k , then the composition moves an object from i to k . For example, ± 1 2 3 2 3 1 ²± 1 2 3 1 3 2 ² D ± 1 2 3 2 1 3 ² : Recall our convention that the element on the right in the product is the first permutation and that on the left is the second. Given this bookkeeping system, you can now write out the multiplication table for the six permu- tations of three objects (Exercise 1.5.1 .) We can match up permutations of three objects with symmetries of an equilateral triangle, so that the multi-
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Unformatted text preview: plication tables match up as well; see Exercise 1.5.2 . Notice that the order in which permutations are multiplied matters in general. For example, 1 2 3 2 1 3 1 2 3 2 3 1 D 1 2 3 1 3 2 ; but 1 2 3 2 3 1 1 2 3 2 1 3 D 1 2 3 3 2 1 : For any natural number n , the permutations of n identical objects can be denoted by two-line arrays of numbers, containing the numbers 1 through n in each row. If a permutation moves an object from position i to position j , then the corresponding two line array has j positioned be-low i . The numbers in the rst row are generally arranged in increasing order, but this is not essential. Permutations are multiplied or composed according to the same rule as given previously for permutations of three objects. For example, we have the following product of permutations of seven objects:...
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