Transpositions, even and odd
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Exercise:
If sigma ...sigma n is obtained from ...n in two different ways one time by a sequence of m transpositions and another times by a sequence of m transpositions and another time by a sequence of m' transpositions then m' and m have the same parity i.e. m' even iff m even and m' odd iff m odd. In other words -^m'-^m. So the parity of m deps only on sigma and not on the particular way we performed a sequence of transpositions ...nrightarrow ...rightarrow sigma ...sigma n. A permutation sigma is called even if m even it is called odd if m odd. The number -^m in - is called the sign of sigma sgnsigma : -^m.
Solution:
Proof. We know that determinant functions M_ntimes nKlongrightarrow K exists forall ngeq . Fix a determinant function E:M_ntimes nKlongrightarrow K we denote it by E to distinguish it from D in the previous discussion. Consider Eepsilon_sigma ...epsilon_sigma n. If sigma_...sigma_n can be obtained from ...n by a sequence of m transpositions then the matrix pmatrix hdots & epsilon_sigma & hdots hdots & vdots & hdots hdots & epsilon_sigma n & hdots pmatrix can be obtained from I pmatrix hdots & epsilon_ & hdots hdots & vdots & hdots hdots & epsilon_n & hdots pmatrix by a sequence of m erchanges of pairs of rows. Longrightarrow Depsilon_sigma ...epsilon_sigma n -^m'Depsilon_...epsilon_n -^m But this applies also to m'. So Depsilon_sigma ...epsilon_sigma n -^m'Depsilon_...epsilon_n -^m'DI-^m' &Longrightarrow -^m' -^m
If sigma ...sigma n is obtained from ...n in two different ways one time by a sequence of m transpositions and another times by a sequence of m transpositions and another time by a sequence of m' transpositions then m' and m have the same parity i.e. m' even iff m even and m' odd iff m odd. In other words -^m'-^m. So the parity of m deps only on sigma and not on the particular way we performed a sequence of transpositions ...nrightarrow ...rightarrow sigma ...sigma n. A permutation sigma is called even if m even it is called odd if m odd. The number -^m in - is called the sign of sigma sgnsigma : -^m.
Solution:
Proof. We know that determinant functions M_ntimes nKlongrightarrow K exists forall ngeq . Fix a determinant function E:M_ntimes nKlongrightarrow K we denote it by E to distinguish it from D in the previous discussion. Consider Eepsilon_sigma ...epsilon_sigma n. If sigma_...sigma_n can be obtained from ...n by a sequence of m transpositions then the matrix pmatrix hdots & epsilon_sigma & hdots hdots & vdots & hdots hdots & epsilon_sigma n & hdots pmatrix can be obtained from I pmatrix hdots & epsilon_ & hdots hdots & vdots & hdots hdots & epsilon_n & hdots pmatrix by a sequence of m erchanges of pairs of rows. Longrightarrow Depsilon_sigma ...epsilon_sigma n -^m'Depsilon_...epsilon_n -^m But this applies also to m'. So Depsilon_sigma ...epsilon_sigma n -^m'Depsilon_...epsilon_n -^m'DI-^m' &Longrightarrow -^m' -^m