Espaces et sous-espaces -

RnRn
{x|xi>0},{x|x1= 0},{x|x1x2= 0},{x|x1+. . . +xn= 0},
{x|x1+. . . +xn= 1},{x|Ax =b,A6=0,b6=0}.
Mn(R)
{ },{ },{ },
{A|BA =AB},{A|A2=A},{A|trace(A)=0}.
R3
{(1,0,0),(0,1,0)},{(1,0,0),(1,1,0),(0,1,1),(1,1,1)},
{(2,1,2),(2,3,5),(0,1,3)},{(2,1,2),(2,3,5),(0,2,3)}.
F G E
F+G F ∩G E
F∪G
F⊂G G ⊂F
Vect(F∪G) = F+G
F G F +G F ∩G
dim(F+G) = dim F+ dim G−dim(F∩G).
ERC∞R R
F={f1, . . . , fn}E x ∈R
W(x) = 
f1(x). . . fn(x)
f0
1(x). . . f0
n(x)
.
.
..
.
.
f(n−1)
1(x). . . f(n−1)
n(x)
.
x0W(x0)F
{t7→ tn|n= 0, . . . , k},{t7→ sin t, t 7→ cos t, t 7→ tsin t},
{t7→ tnet|n= 0, . . . , k},{t7→ sin(nt)|n∈N∗},{t7→ |t−a| | a∈R}.
j1j2
1j2j
j2j1
,
0. . . 0t1
.
.
.....
.
..
.
.
0. . . 0tn−1
t1. . . tn−1tn
,
1a0. . . . . . 0
0 1 a....
.
.
.
.
..............
.
.
.
.
...........
.
.
0...1a
a0. . . . . . 0 1
.

rMp,q(K)Ir0
0 0 Mp,q(K)
A∈ Mn,p(K)B∈ Mp,m(K)
rang(AB)6inf(rang(A),rang(B)).
A,B∈ Mp,q(K)
rang(A+B)6rang(A) + rang(B).
A 0
0 B A B
A∈ Mm,n(K) 1 x∈Kmy∈Kn
A=xy>
A∈ Mm,n(K)rB∈ Mm,r (K)C∈ Mr,n(K)
rA=BC
A∈ Mn(K)
A−1=1
det(A)(ComA)>,
ComA A
E I R
f g E
x0, x1∈I
f(x0)g(x1)−f(x1)g(x0)6= 0.
{t7→ cos t, t 7→ sin t},{t7→ eat, t 7→ teat}, a ∈R.
(x0, y0) (x1, y1) (x2, y2)
1
2
x1−x0x2−x0
y1−y0y2−y1
.
A,B∈ Mn(K)
det(AB) = det(A)det(B).
A D
det A B
0 D = det(A)det(D).
det A B
C D A,B,C,D∈ Mn(K)A

n>2
Vn=
1x1x2
1. . . xn−1
1
1x2x2
2. . . xn−1
2
.
.
..
.
..
.
..
.
.
1xnx2
n. . . xn−1 n
.
Vn
a1+a2x+a3x2+. . . +anxn−1= 0,
n>2ai∈Rn ai= 0 i
R
x+y= 4
3y+ 4z= 1
2x+ 2y= 8
,
x+y= 3
−x+ 5y+ 6z= 2
2x+ 3y+z= 9
,
ax +y+z=b
x+ay +z=c, a, b, c ∈R.
A=
1 2
3 4
−1 4
B BA
A B
DA∈ Mp(K)D∈ Mq(K)A
D
1 1 1 1
1 1 −1−1
1−1 1 −1
1−1−1 1
FR4
(1,−1,2,3),(1,1,2,0),(3,−1,6,−6),
GR4
(0,2,0,−3),(1,0,1,0).
F G F ∩G F +G
Eu −→ Fv −→ G v ◦u= 0
Im u⊆Ker v
u v KE u ◦v=v◦u
u+v= idE
Ker (u◦v) = Ker u⊕Ker v.
uKE
E= Ker u⊕Im uKer u= Ker u2
Im u= Ker u u2= 0 dim E= 2.rang u

uKE
E= Ker u⊕Im uKer u= Ker u2Im u= Im u2.
(Ker un)n∈N,(Im un)n∈N.
n
Ker un⊆Ker un+1,Im un+1 ⊆Im un.
NKer uN= Ker uN+1
Ker un= Ker uN, n >N.
NIm uN= Im uN+1
Im un= Im uN, n >N.
n > 0
Ker un= Ker un+1 Ker un∩Im un={0}.
Im un= Im un+1 E= Ker un+ Im un.
E
K
trace : Mn(K)−→ K
AMn(K) ΓA:Mn(K)→K
ΓA(M) = trace(AM)
Mn(K)
ϕMn(K)Aϕ= ΓA
ϕMn(K)ϕ(MN) = ϕ(NM)λ∈K
ϕ(M) = λtrace(M)M∈ Mn(K)
EKF E F
F⊥={ϕ∈E∗|ϕ(x) = 0 x∈F}.
F⊥E∗
{e1, . . . , ep}F
F⊥={ϕ∈E∗|ϕ(e1) = . . . =ϕ(ep)=0}.
E
dim E= dim F+ dim F⊥.
G E
(F⊥)⊥=F;G⊂F F ⊥⊂G⊥;
(F+G)⊥=F⊥∩G⊥; (F∩G)⊥=F⊥+G⊥.

A=
1a2a2a
1 1 a1
1a1 1
a a2a21
.
A−(1 −a)141−aA
Aa= 0
aA
010. . . 0
001 ....
.
.
000 ...0
.
.
..
.
..
.
....1
000. . . 0
∈ Mn(R),
0 1 0 . . . 0
.
.
.0 1 ....
.
.
.
.
.......0
0...1
1 0 . . . . . . 0
∈ Mn(C),
n1 1 . . . 1
1n1. . . 1
1 1 n . . . 1
.
.
..
.
..
.
.....
.
.
1 1 1 . . . n
∈ Mn(R),
1. . . 1−n
.
.
..
.
..
.
.
1. . . 1−n
∈ Mn+1(R),
1 0 0 1
0... ...
1 1
1 1
0... ...0
1 0 0 1
∈ M2p(R).
Mn(R)n>2
A=
a b . . . b
b a ....
.
.
.
.
.......b
b . . . b a
.
A A
a b A
A
Akk eA
uRn
u(P)=(X2−1)P00 + 3XP 0,
A∈ Mn(K) rang(A) = 1
Atrace(A)6= 0.
u v Ku◦v=v◦u
u v
u v