Permutations, the Parity Theorem, and Determinants
نویسنده
چکیده
The Parity Theorem says that whenever an even (resp. odd) permutation is expressed as a composition of transpositions, the number of transpositions must be even (resp. odd). The purpose of this article is to give a simple definition of when a permutation is even or odd, and develop just enough background to prove the parity theorem. Several examples are included to illustrate the use of the notation and concepts as they are introduced. We then define the determinant in terms of the parity of permutations. We establish basic properties of the determinant. In particular, we show that detBA = detB detA, and we show that A is nonsingular if and only if detA 6= 0. If you find this writeup useful, or if you find typos or mistakes, please let me know at [email protected] 1. What is a Permutation A permutation is an invertible function that maps a finite set to itself.1 If we specify an order for the elements in the finite set and apply a given permutation to each point in order, then the function values we generate simply list all the points 1 To say that a function is invertible means that it is both one-to-one and onto. One-to-one means that no pair of points can map to a common destination point. Onto means that every point is the image of some point.
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تاریخ انتشار 2014