Notions d`Algèbre Linéaire

Mm,n(K)

K R C
EK+. E
E E
K
+ : E×E→E
(x, y)7→ x+y,.:K×E→E (x, λ)7→ λ.x
(E, +, .)K K
•(E, +)
+E∀x, y ∈E x +y=y+x
+∀x, y, z ∈E x + (y+z)=(x+y) + z
E0E∀x∈E x + 0E=x
x E + (−x)x+ (−x) = 0E
•.
∀λ, µ ∈K∀x∈E, (λ+µ).x =λ.x +µ.x,
∀λ∈K∀x, y ∈E λ.(x+y) = λ.x +λ.y
∀λ, µ ∈K(λµ).x =λ.(µ.x)
∀x∈E, 1K.x =x
KE
KE
(λµ).x λµ
K.Kλµ x E +
KE
RnR+. x, y ∈Rn, λ ∈R
x+y:= (x1+y1, ..., xn+yn)
λ.x := (λx1, ..., λxn)
C0(I,R) = { I ⊂ R R}R+.
f, g ∈E, λ ∈R
∀x∈I, (f+g)(x) := f(x) + g(x)∀x∈I, (λ.f)(x) := λf(x)
Kλ∈Kx∈E
λ.x = 0E⇔(λ= 0Kx= 0E)

(⇐)λ.0E=λ.(0E+ 0E)⇔λ.0E=λ.0E+λ.0Eλ.0E+ (−(λ.0E))
| {z }
0E
=λ.0E+
λ.0E+ (−(λ.0E))
| {z }
0E
0E=λ.0E+ 0E0E=λ.0E
(⇒)λ∈Kx∈E λ.x = 0E. λ 6= 0Kx= (λ−1λ).x =λ−1.(λ.x) =
λ−1.0E= 0E.
KE F ⊂E F
E
F+∀x, y ∈F x +y∈F
F . ∀λ∈K∀x∈F λ.x ∈F
F∀λ∈K∀x, y ∈F
λ.x +y∈F.
R×{0}R2.{(x, λ.x), λ ∈K}
R2.
C1(I,R) = { I ⊂ R R} C0(I,R)
(F, +)
I+F+E
IF+ (−1).x +x= ((−1) + 1).x = 0E∈F0E
+F
I∀x∈F(−1).x ∈F x +−x
. . E
EKx1, ..., xpE x1, ..., xp
λ1.x1+... +λp.xpλi∈Ki= 1 : p.
p
P
i=1
λi.xi
E=R4, x1, x2, x3R4. y = 3.x1−2.x3
x1, x2, x3

(x1, ..., xp)E
∀λ1, ..., λpK
p
X
i=1
λi.xi= 0E⇒λi= 0Ki=1:p.
∃λ1, ..., λpK
p
X
i=1
λi.xi= 0E.
E=R2x1= (1,2) x2= (−3,−6) 3.x1+x2= 0.
x3= (−1,3) (x1, x3)∀λ1, λ3K
λ1.x1+λ3.x3= 0R2⇔λ1−λ3= 0
2λ1+ 3λ3= 0 ⇔λ1= 0 λ3= 0
EKA E
A E A
V ect(A). A
V ect(A) = E A E
IV ect(A)E V ect(A)
0Eai∈A
V ect(A)⊂E
A E E
V ect(A)λ∈Ket x, y ∈E
λ.x +y=λ.
r1
X
i=1
λi.ai+
r2
X
j=1
µj.aj=
r1
X
i=1
(λλi).ai+
r2
X
j=1
µj.aj.
A:
λ.x +y=
r3
X
i=1
νi.ai,
λ.x +y∈V ect(A)
IH A ⊂H V ect(A)⊂H x ∈V ect(A)x
A H H x ∈H.
A V ect(A)V ect(A)
A.