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Chapitre3

K R C

E K n>1 ȱ B=

(e1,...,en) f:En-→K f n fn xi f(x1,...,xn) = 0ȱ f(e1,...,en) = 1 detB ȱ ȱ B

ȱdetB (y1,...,yn)

xi detB(y1,...,yn) =-detB(x1,...,xn). detB(x1,...,xi-1,xi+xj,xi+1...,xj-1,xj+xi,xj+1,...,xn) = 0 ij λ detB(x1,...,xi,...,xj,...,xn) = detB(x1,...,xi+λxj,...,xj,...,xn).

ȱ j

f:En-→K n

λ f=λdetB λ=f(e1,...,en) ȱ1

λf detB f ȱ

B?= (e?1,...,e?n) E

detB?= detB?(e1,...,en)detB.

λdetB?=λdetB (e1,...,en)

(x1,...,xn) n detB(x1,...,xn)?= 0 B? ȭ ȱ detB= detB(x1,...,xn)detB? detB(x1,...,xn)?= 0

ȱdetBȱ ȱ

ȭ ȱ ȱ xj xj=?

i?=jλixi xj←x?j=xj-? i?=jλixi ȱ x?jȱ

BE ȱ

ȱ (x1,...,xn) ȱ

E

B= (e1,...,en)B?= (e?1,...,e?n) E f E

detB?f(e1),...,f(en)?= detB??f(e?1),...,f(e?n)?.

ȱ E ȱf

det(f)

λ det(λIdE) =λn B= (e1,...,en) E

det(λIdE) = detB(λe1,...,λen) =λndetB(e1,...,en) det(IdE) = 1

ȱ det(λIdE) =λdet(IdE) =λ ȱ

det(f+g)

B= (e1,...,en) E ?f(e1),...,f(en)? E

f fE det(f)?= 0 f,g E det(g◦f) = det(g)det(f). f E det(f-1) =1 det(f). det(f)det(f-1) = det(f◦f-1) = det(IdE) = 1

λ fE

det(λf) =λndet(f).

λf= (λIdE)◦f

M?Mn(K) M

ȱf?(Kn)

B= (e1,...,en) KnM1,...,Mn M

m1,...,mn Kn matB(mk) =Mk det(M) = detB(m1,...,mn). f ȱ E BE det(f) = det?matB(f)?.

B= (e1,...,en) E

det(f) = detB?f(e1),...,f(en)?= det matB?f(e1),...,f(en)?? = det?matB(f)?.

ȭ M det(M) = 0

ȭ M? M det(M?) =-det(M)

ȭ M? M det(M?) = det(M)

ȭ det(n) = 1

M= [mi,j]i,j?[[1,n]]

det(M) = n? k=1 mk,k. det(M) = 0

ȭ λ M?Mn(K) det(λM) =λndet(M)

ȭ M det(M)?= 0 ȱ

det(M-1) =1 det(M);

ȭ M,N?Mn(K) det(MN) = det(M)det(N)

MN det(M) = det(N)

P?n(K)N=P-1MP

det(N) = det(P-1MP) = det(P-1)det(M)det(P) =1 det(P)det(M)det(P) = det(M).

M= [1 00 1]N= [1 10 1]

P?2(K)

P-1MP=P-12P=2?=N

Mȱ ȱ

????a b c d ????=ad-bc a1,1a1,2a1,3 a2,1a2,2a2,3 a3,1a3,2a3,3

ȱ Mn(K)n>4

a11a12a13 a21a22a23 a31a32a33 a11a12a13 a21a22a23

M?Mn(K) det(M?) = det(M)

A?Mn(K)n>2

ȭj?[[1,n]]

det(A) = n? i=1 (-1)i+jai,jDi,j(A) j

ȭi?[[1,n]]

det(A) = n? j=1 (-1)i+jai,jDi,j(A) i 2 8 1 6 3 5 7 4 9 2 8 1 6 3 5 7 4 9 2 ??????+ 5 8 1 6 3 5 7 4 9 2 ??????-9 8 1 6 3 5 7 4 9 2 ????3 7 4 2 ????+ 5 ????8 6 4 2 ????-9 ????8 6 3 7

A?Mn(K)B?Mn,p(K)C?Mp(K)

det ?A B 0C = det(A)det(C). ?nB 0C ??A0 0p ?A B 0C n det ?nB 0C = det(C), det ?A0 0p = det(A) p ȱ a1,...,an K

1ana2n···an-1n

16i (aj-ai).

Vn(a1,...,an) Cn←Cn-a1Cn-1

Cn-1←Cn-1-a1Cn-2 C2←C2-a1C1

Vn(a1,...,an) =

1 0 0···0

1a2-a1a2(a2-a1)···an-22(a2-a1)

1an-a1an(an-a1)···an-2n(an-a1)

a2-a1 a3-a1 Vn(a1,...,an) = (an-a1)···(a2-a1)Vn-1(a2,...,an).

ȭ n= 2 ??1a11a2

??=a2-a1

ȭ n-1 ȱ ȱ

n Vn(a1,...,an) = (an-a1)···(a2-a1)Vn-1(a2,...,an) = (an-a1)···(a2-a1)?

26i (aj-ai) =?

16i (aj-ai). ai fȱ f:Kn-1[X]-→Kn

P?-→?P(a1),...,P(an)?,

f ȱ ȱKn-1[X]

ȱKn

ȭ ai an=an-1 ȱ ȱ

P= (X-a1)···(X-an-1) ȱ

ȭ ai ȱ

n-1 n

ER n>1E ȱ

E

B1,B2 EPB1,B2 B1 B2

B2

ȭ B1 det(PB1,B2)>0

ȭȱ B1 det(PB1,B2)<0

B2

B1 B1

B2 PB1,B2PB2,B1 ȱ ȱ

ȱ E

BE CE

B BCdiag(-1,1,...,1)

-1 ȱ DE

B C ȱ

PB,D=PB,CPC,D,

det(PB,D) = det(PB,C)det(PC,D).

ȭD B det(PB,D)>0 det(PC,D) =-det(pB,D)<0D

ȱ C

ȭD ȱ D det(PB,D)<0 det(PC,D) =-det(pB,D)>0D C

ȱE ȱ

E ȱ

B= (e1,...,en) EB?= (e2,...,en,e1) ȱ

B B B?

P=

0 0···0 1

1 0···0 0

0 1 0 0

0···0 1 0

n-1C1↔C2 C2↔C3 ȹCn-1↔Cn n det(P) = (-1)n-1

ȭn n= 2 BB? ȱ

ȭn n= 3 BB?

B1B2 ȱ B2B3

B1B3 det(PB1,B3) = det(PB1,B2)det(PB2,B3)>0. Cquotesdbs_dbs50.pdfusesText_50

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