Algèbre linéaire: généralités 1. E = C(R,R) . Pour tout entier n, on dé

2014 −2015
E=C(R,R)n fn:x7→ cos(2x+n)
N(f0, ..., fN)
F1F2E F1∪F2
F1⊂F2F2⊂F1F1∩F2=F1+F2F1=F2
E=F(R,C)
ep:x7→ exp(ipx)p∈Z
E1={(x, y, z, t)∈R4/−x−2y+ 2z+t= 0}D=V ect((1,2,1,1)) E1
D
F1={(x, y, z, t)∈R4/2x−y+z=y−2z+t= 0}
F2={(x, y, z, t)∈R4/x + 2y−t=x+ 2z−t= 0}
F1F2
R[X]B0= 1
Bp=X(X−1)...(X−p+ 1)
(Bp)p∈NR[X]
Z
E=C([0,2],R)F E f E
F
E n n > 0 (e1, ..., en)E
(λ1, λ2, ..., λn)Kni i = 1..n ui=u+ei
u u =
n
∑
i=1
λiei
(u1, ..., un)E
n
∑
i=1
λ1̸=−1
E, F Kf∈ L(E, F )G E
ker f˜ f f G ˜ f G
Im f
E f ∈L(E)Im(f)
f Im(f)∩Ker(f) = {−→ 0E}
Kerf Imf
f
E f ∈L(E)
(i) : f2=O
(ii) : ∃(g, h)∈(L(E))2/g ◦h=f h ◦g=O
E K n > 0f1, ..., fnn
x∈E∀i∈ {1, ..., n}fi(x) = 0
n

2014 −2015
E∗f g ∀x∈E
f(x)g(x) = 0
E={ }
i∈Nφi:E→R
(un)n∈N7→ uiφi∈E∗
(φi)i∈N
ψ:E→R
(un)n∈N7→ lim unψ
φi
E=R3P1: 3x−2y+z= 0 P2:x+ 3y−z= 0 D=P1∩P2
P
D
E=R3[X]a b φ1:
P7→ P(a)φ1:P7→ P′(a)φ1:P7→ P(b)φ1:P7→ P′(b)
E∗
(e1, e2, e3, e4)E i j
φi(ej) = δi,j
E n F, G E Φ
u∈ L(E) (u/F, u/G)L(F, E)× L(G, E)
Φ
E=R[X]L P E
L(P) = P(X+ 1) + P(X−1) −2P(X)L
L(P) deg(P) = n > 1
L L
E, F, G K u ∈ L(E, F )v∈ L(F, G)
rg(v◦u)≤inf(rg(u), rg(v))
v′=v/u(E)rg(u) + rg(v)−dim(F)≤rg(v◦u)
p n p ≤n f ∈ L(Rn,Rp)g∈
L(Rp,Rn)f◦g=IdRpg◦f
E F f ∈ L(E, F )
f g ∈
L(F, E)rg(f◦g) = rg(g◦f)rg
g∈ L(F, E)rg(f◦g) = rg(g◦f)
f̸= 0 x0∈E f(x0)̸= 0
Im(f)̸=F g ∈ L(F, E)g◦f= 0
f◦g̸= 0 f
rg(f◦g) = rg(g)g∈ L(F, E)
f