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lecture18 - Lecture 18 We begin with a quick review of...

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�� Lecture 18 We begin with a quick review of permutations (from last lecture). A permutation of order k is a bijective map σ : { 1 , . . . , k } { 1 , . . . , k } . We denote by S k the set of permutations of order k . The set S k has some nice properties. If σ S k , then σ 1 S k . The inverse permutation σ 1 is defined by σ 1 ( j ) = i if σ ( i ) = j . Another nice property is that if σ, τ S k , then στ S k , where στ ( i ) = σ ( τ ( i )). That is, if τ ( i ) = j and σ ( j ) = k , then στ ( i ) = k . Take 1 i < j k , and define τ i,j ( i ) = j (4.40) τ i,j ( j ) = i (4.41) τ i,j ( ) = �, � = i, j. (4.42) The permutation τ i,j is a transposition. It is an elementary transposition of j = i +1. Last time we stated the following theorem. Theorem 4.13. Every permutation σ can be written as a product σ = τ 1 τ 2 τ r , (4.43) · · · where the τ i ’s are elementary transpositions. In the above, we removed the symbol denoting composition of permutations, but the composition is still implied. Last time we also defined the sign of a permutation Definition 4.14. The sign of a permutation σ is ( 1) σ = ( 1) r , where r is as in the above theorem. Theorem 4.15. The above definition of sign is well-defined, and ( 1) στ = ( 1) σ ( 1) τ . (4.44) All of the above is discussed in the Multi-linear Algebra Notes. Part of today’s homework is to show the following two statements: 1. | S k | = k !. The proof is by induction. 2. ( 1) τ i,j = 1 . Hint: use induction and τ i,j = ( τ j 1 ,j )( τ i,j 1 )( τ j 1 ,j ), with i < j .
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