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E
EK=R C
x, y, a, u ~x, ~y, ~a, ~u
α, β, λ, µ

A E ~x ∈E
~x A (−→ a1,−→ a2,...,−→ ak)A k
α1, α2, . . . , αk
~x =α1−→ a1+α2−→ a2+· · · +αk−→ ak
E=M2(K), M =a b
c d , I2=1 0 0 1 
M2I2M
ad −bc 6= 0 M M I2
Vect A A
Vect ∅=−→ 0E
Vect A E A
Vect A E Vect A A
AVect A
Vect A E
Vect A−→ 0E∈Vect A−→ x A −→ 0E0−→ x
A
Vect A A
AVect A
−→
x∈Vect A λ ∈Kλ−→ x A Vect A
Vect A A −→ x∈A−→ x= 1−→ x∈Vect A
F A Vect A−→ xVect A−→ x=
n
X
i=1
αi−→ aiαi
K−→ aiA A ⊂F−→ aiF−→ x=
n
X
i=1
αi−→ ai
F F E
Vect A⊂F
F E A F F = Vect A A F
A F
(−→ ai)i∈I

(−→ x1,−→ x2,...,−→ xn)
∀α1, α2, . . . , αn∈K,α1−→ x1+α2−→ x2+· · · +αn−→ xn=−→ 0E=⇒α1=α2=· · · =αn= 0
(−→ x1,−→ x2,...,−→ xn)α1, α2, . . . , αn∈K
α1−→ x1+α2−→ x2+· · · +αn−→ xn=−→ 0E
P1, P2, . . . , Pn∈K[X]\ {0}deg P1<deg P2<· · · <deg Pn(P1, P2, . . . , Pn)
(−→ xi)i∈I
I⊂N(Pi)i∈I
E=F(R∗
+,R) (fa)a∈Rfa:x7→ | ln x|a
E=RR(gα)α∈Rgα:x7→ sin(x+α)
(−→ x1,−→ x2,...,−→ xn) (−→ x1,−→ x2,...,−−−→ xn−1)−→ xn
−→
x1,−→ x2,...,−−−→ xn−1
(−→ x1,−→ x2,...,−→ xn)α1, α2,...,αn∈K
α1−→ x1+α2−→ x2+···+αn−→ xn=−→ 0E
αn
α1−→ x1+α2−→ x2+···+αn−1−−−→ xn−1=−→ 0E
α1, α2,...,αn−1(−→ x1,−→ x2,...,−−−→ xn−1)
αn6= 0
−→
xn−→ x1,−→ x2,...,−−−→ xn−1
−→
xn=
n−1
X
i=1 −αi
αn−→
xi
−→
x−→ y−→ x6=−→ 0λ∈K
−→
y=λ−→ x

EKE= (−→ ei)i∈IE
EE⇐⇒ E
EKE= (−→ ei)i∈IE
EE⇐⇒ 
−→
x E
E
−→
x
E
E
E−→ x
E
E E
E
dim{−→ 0}= 0
E
EKnE n
E n E Kn
E= (−→ e1,−→ e2,...,−→ en)E−→ x E
−→
x=
n
X
i=1
xi−→ eiϕ:−→ x7→ 
x1
x2
xn
EKn
ϕ
ϕ ϕ(−→ x) = 
x1
x2
xn
ϕ(−→ y) = 
y1
y2
yn
ϕ(λ−→ x+−→ y) =

λ
x1
x2
xn
+
y1
y2
yn
ϕ
−→
x:
x1
x2
xn
−→
x
E=KnEKn
dim E=n n n
n n
dim E=n F E 6ndim F=n F =E
F1F2
E
F1⊂F2
dim F1= dim F2=⇒F1=F2
EKn>1f E p
fp= 0
x∈Ex, f(x), f2(x), . . . , fp−1(x)
fn= 0
Kn
1
0
0
0
,
0
1
0
0
,
0
0
1
0
,...,
0
0
0
1
Mpq(K)p q K
Eij i j Mpq(K)
p×q
K[X] (Xk)k∈N
Kn[X]K[X]6n
(Xk)06k6n