corrigé

dim V×W= dim V·dim W
f(x) + f(−x)
2,f(x)−f(−x) 2.
A+AT
2,A−AT 2.
b= 0
λ6=±1
f:R3−→ R4
f(x, y, z) = (2x−2y+ 4z, 5x−4y+ 7z, 3x−2y+ 3z, x −y+ 2z).

2−2 4
5−4 7
3−2 3
1−1 2
{(1,3,1)}
{(2,5,3,1),(−2,−4,−2,−1)}
V W K f :V−→ W
b= (−16,23,0,0,0)
W=(1,−2,1,0,0),(1,−2,0,1,0),(5,−6,0,0,1).
b+w w ∈W
a= 0 b6=−1
a= 0 b=−1
f1(x) = cos x, f2(x) = sin x C0(R,R)V
f1f2
0 1
−1 0 
B=∅0
B={f1, f2}
A−1=
3−5 3
−1 3 −2
0−1 1
.
det A= (c−a)(b−a)(c−b)

A=
1 1 −b0
0b0
1 + b1 + b b
,
b∈R
pA(t) = (b−t)2(1 −t) 1 b
b= 1 1
2{(1,−1,0),(0,0,1)}
b6= 1 1 1
1 (1 −b, 0,1 + b)
b2
{(1,−1,0),(0,0,1)}
pA(t) = (t−2)2·(6 −t)
A=
1 0 0
0 cos α−sin α
0 sin αcos α
,
α6=kπ k ∈Z
λ= 1 λ= cos α+isin α λ = cos α−isin α
F
VRT:V−→ V
(T−idV)2◦(T+ 2 ·idV)6= 0,(T−idV)3◦(T+ 2 ·idV)=0.
(t−2)2(t−2)2·(t+ 2)
S=
−1 0 −2
010
101

λ=−10 λ= 20
(1,−2,−1,1) (1,1,1,1)
(10 ·E4−A)2=
−10 80 140 −110
−170 160 380 −270
−10 80 140 −110
310 −80 −340 210
.
10
(2,−5,3,0),(−1,4,0,3).
VKn≥1F:V→V
λ∈K
A∈M(n×n, K)Kn≥1A
VKn≥1F:V→V F
λ∈K
A∈M(3 ×3,R)A100 = 0
F:C8→C8pF= (t−2) ·(t+ 5)4·(t−10)3MF=
(t−2) ·(t+ 5)2·(t−10)2
F