Ma1bAnNotes3 - Chapter 3 Determinants Let F = R or C and Mn...

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Chapter 3. Determinants. Let F = R or C and M n the set of all n by n matrices over F . The text defines the determinant as a function from M n into F satisfying certain axioms. It goes on to derive various properties of such a function and to show that there exists at most one such function; ie. it proves a uniqueness result. Finally in section 3.13 it shows such a function exists, but does not write down the function explicitly. Our approach will be different. We will take the standard approach of defining an explicit function det : M n F and proving the function det satisfies the axioms in the text and has various other useful properties. We will not worry about uniqueness. Permutations. To define the determinant we some facts about permutations. Let I = { 1 ,... ,n } . A permutation of I is a 1-1 correspondence s : I I of I with itself. Write S n for the set of all permutations of I . In the literature S n is called the symmetric group of degree n . Cycle Notation. We need some notation to describe permutations. Each s S n can be written in cycle notation : s = ( a 1 ,... ,a α )( b 1 ,... ,b β ) ··· ( z 1 ,... ,z ζ ) . In this notation the entries a i ,b i ,... ,z i are the members of I in some order. The notation indicates that s ( a i ) = a i +1 for 1 i < α and s ( a α ) = a 1 . Similarly s ( z i ) = z i +1 for i < ζ and s ( z ζ ) = z 1 . Example 1. Let n = 6. Then r = (2 , 1 , 4)(5 , 3 , 6) and t = (1)(2 , 6)(3)(4)(5) are the members of S 6 such that 2 r -→ 1 r -→ 4 r -→ 2 , 5 r -→ 3 r -→ 6 r -→ 5 and such that t fixes 1,3,4, and 5 and interchanges 2 and 6. 1
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2 The term ( a 1 ,... ,a α ) is called a cycle of s and this particular cycle is said to be of length α as it involves α members of I . Thus in Example 1, r has two cycles of length 3 and t has four cycles of length 1 and one cycle of length 2. Define a permutation s to be even if s has an even number of cycles of even length, and define s to be odd if s has an odd number of cycles of even length. Thus in Example 1, r is even since it has zero cycles of even length, and t is odd as it has one cycle of even length. A permutation like t with one cycle of length 2 and n - 2 fixed points is called a transposition . Usually by convention we suppress cycles of length 1. Subject to this convention, we write the transposition t as (2 , 6) rather than (1)(2 , 6)(3)(4)(5). Define the sign function sgn : S n → {± 1 } on S n by sgn( s ) = +1 if s is even and sgn( s ) = - 1 if s is odd. We now define a function det : M n F called the determinant function. Given a matrix A = ( a i,j ) M n define the determinant of A to be det( A ) = X s S n sgn( s ) · a 1 ,s (1) a 2 ,s (2) ··· a n,s ( n ) . In order to talk about the determinant, we establish some notation: Given
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This note was uploaded on 04/28/2010 for the course MATH 1B taught by Professor Aschbacher during the Winter '07 term at Caltech.

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Ma1bAnNotes3 - Chapter 3 Determinants Let F = R or C and Mn...

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