2011 S1 UE Renforcements Mathématiques Algèbre linéaire

2010 −2011
S1
n E
R Rn0EE IdE
E
F E
f E f F f(F)F
E E
E
R[X]
kRk[X]
R[X]k f E
f0=IdE, f1=f, f2=f◦f
P=
n
k=0
akXk
R[X]P(f)
E
P(f) =
n
k=0
akfk.
f E
PR[X]
P(f)
f

E f
f
B= (e1, e2, e3)R3
gR3
B
110
010
002
.
R3g
p
F1, ..., FpE
f
p
k=1
Fk
E f
λ1, ..., λpf n1, ..., np
p
k=1
(f−λkIdE)nk
f
λ
E f
(f−λIdE)
f f2
f
E
f
f−1
f
E
R2
R2

E
ERE E
n−1
E
E
ϕ E H =ϕ
H f
λR
ϕ◦f=λϕ.
A f
E L ϕ
ER
H f
tAtL=λtL.
R3
g
f E
λ1, ..., λpE1, ..., Ep
f
p= 1
p2
F E f x
F
(x1, x2, ..., xp)∈
Πp
k=1Ek
x=
p
k=1
xk.
p
k=2
(λk−λ1)xk
F

x1, x2, ..., xpF
E f
p
k=1
Fk
k1≤k≤p Fk
Ek
f
F
f E
f
n
DRn−1[X]
P′
DnDn−1
Rn−1[X]
D
n
R0[X],R1[X], ..., Rn−1[X].
0E
f E n
fn= 0 fn−1̸= 0
B= (e1, e2, ..., en)E
A=
010. . 0
0010. .
. . . . . .
. . . . 0 1
. . . . . 0
,
A i
j1j=i+ 1 0

A B
B=
010. . 0
0020. .
. . . . . .
. . . . 0n−1
. . . . . 0
,
B i
j i j =i+1 0
E
f
2
f E
2E f ◦f
F2
F2∩f={0}.
f⊇f(F2)
F1f
f(F2)F1+F2
E f
A, B, C E
(A+B)∩C⊇(A∩C)+(B∩C).
(F1 + F2) ∩f
F
E f F1=F∩f
F2F1F
f⊆f(F)F2
f
n= 4 E=R4
h E