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(λ1, ...λn)∈Knµ∈K
n
X
k=1
λiui+µv = 0.
µ
v=
n
X
k=1
−λk
µuk∈Vect(u1, ..., un),
µ= 0
n
X
i=1
λiui= 0,
u1, ..., un
λ1=λ2=... =λn= 0.
u1, ..., un, v
u∈Vect(v1, v2)
∃(λ1, λ2)∈K2, u =λ1v1+λ2v2.
u= (λ1−λ2)v1+λ2(v1+v2)∈Vect(v1, v1+v2).
Vect(v1, v2)∈Vect(v1, v1+v2)u∈Vect(v1, v1+v2)
∃(λ1, λ2)∈K2, u =λ1v1+λ2(v1+v2).
u= (λ1+λ2)v1+λ2v2∈Vect(v1, v2).
Vect(v1, v1+v2)⊂Vect(v1, v2)

u v KE
x∈ker(u)∩ker(v)
u(x) = v(x) = 0 (u+v)(x) = u(x) + v(x) = 0 + 0 = 0,
x∈ker(u+v) ker(u)∩ker(v)⊂ker(u+v)
u
E v =−u
ker(v) = ker(u) = {0} ⇒ ker(u)∩ker(v) = {0},
ker(u+v) = ker(0) = E
y∈Im(u+v)
∃x∈E, y = (u+v)(x) = u(x) + v(x)∈Im(u) + Im(v).
Im(u+v)⊂Im(u) + Im(v).
u
E v =−u
Im(u+v) = Im(0) = {0} 6⊂ E= Im(u) + Im(v).
x∈ker u
u2(x) = u(u(x)) = u(0) = 0,
x∈ker(u2) ker(u)⊂ker(u2).
E=R2u
0 1
0 0,
u6= 0 u2= 0
E= ker(u2)6⊂ ker(u).
yIm(u2)
∃x∈E, y =u2(x) = u(u(x)) ∈Im(u),
Im(u2)⊂Im(u)
E=R2
u
0 1
0 0,
u6= 0 u2= 0
Im(u)6⊂ {0}= Im(u2).

u v
y∈Im(u)
∃x∈E, y =u(x),
v(y) = v(u(x)) = v◦u(x) = u◦v(x) = u(v(x)) ∈Im(u),
Im(u)v
x∈ker(u)
0 = v(0) = v(u(x)) = v◦u(x) = u◦v(x) = u(v(x)),
v(x)∈ker(u) ker(u)v
y∈Im(u)
∃x∈E, y =u(x).
∃(a, b)∈ker(u)×ker(v), x =a+b.
v(y) = v(u(x)) = v◦u(x) = v◦u(a+b) = v◦u(a)+v◦u(b) = v(u(a))+u◦v(b) = v(0)+u(v(b)) = 0+u(0) = 0.
y∈ker(v)
Im(u)⊂ker(v).
u v
Im(v)⊂ker(u).
E
dim(E) = dim(ker(u)⊕ker(v)) = dim(ker(u)) + dim(ker(v)).
dim(E) = dim(Im(u)) + dim(ker(u)) = dim(Im(v)) + dim(ker(v)).
dim(Im(u) = dim(ker(v)),dim(Im(v)) = dim(ker(u)),
ker(u) = Im(v),ker(v) = Im(u).

f◦pu=pu◦f
f(u) = f◦pu(u) = pu◦f(u) = pu(f(u)∈Vect(u),
u
∀u∈E∗,∃λu∈K, f(u) = λuu.
v u v u +v
λu+v(u+v) = f(u+v) = f(u) + f(v) = λuu+λvv,
(λu+v−λu)u+ (λu+v−λv)v= 0,
(u, v)
λu=λu+v=λv.
µ∈K∗
λµu(µu) = f(µu) = µf (u) = µλuu=λu(µu),
µu 6= 0
λµu =λu.
e E
∀x∈E∗, f(x) = λex,
x= 0 f=λeId
E

E=R2F={(x, y)∈E:x=y}
(1,0) + (0,1) = (1,1) ∈F,
(1,0) 6∈ F(0,1) 6∈ F
0v1+v2+ (−1)v2= 0,0v1+ 0v2+v4= 0.
421
032
001
v3v2v1
E=R2u= (1,0), v = (0,1), w = (1,1) (u, v)
(u, w) (v, w)
u+v−w= 0 ⇒(u, v, w).
E=R3F= Vect(e1)G= Vect(e2)H= Vect(e1, e2)
F+H=H=G+H, F 6=G.
(f1, ..., fn)F
g1, ..., gmG h1, ..., hpH
f1, ..., fn, h1, ..., hpg1, ..., gm, h1, ..., hpE
n+p= dim(E) = m+p⇒n=m.
E
∀i∈J1, nK, u(fi) = gi,
∀j∈J1, pK, u(hj) = hj.
u E u
F G u F G
E=R2F= Vect(e1)G= Vect(e2)
e1, e2∈F∪G, e1+e26∈ F∪G.
x, y ∈Ti∈I(Ei)λ∈K
∀i∈I, x, y ∈Ei⇒x+λy ∈Ei,
x+λy ∈Ti∈I(Ei).
u E
ker(u) = {0},Im(u) = E,
ker(u)⊕Im(u) = E.